Additional buoy type cable tension characteristic analysis method

Abstract

The invention discloses an additional buoy type cable tension characteristic analysis method. The method comprises the following steps: determining additional buoy type cable model parameters and sea condition parameters; parameters of the buoy are set according to the buoyancy-weight ratio; motion of the cable unit is described, and an initial configuration of the cable unit and a global position vector of any point on the axis of the cable unit of a deformed configuration are given based on an absolute node coordinate method; constructing an expression of a cable unit kinetic energy and mass matrix; calculating strain energy and elastic force of the cable unit; calculating an external load borne by the cable unit to obtain a generalized external force matrix of the cable unit; assembling a mass matrix, an absolute node coordinate vector, a generalized external force matrix and a cable elasticity matrix of each unit on the cable; establishing a mooring rope kinetic equation of the additional buoy; and carrying out dynamic solution on the motion equation set in the matrix form. According to the invention, the effect of the buoy in the cable system can be accurately described, so that the simulation precision is remarkably 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 tension characteristics of an additional buoy-type cable. Background Art

[0002] The cable system of an offshore floating structure is crucial for ensuring the stability and safety of the platform. Traditional cable modeling methods mainly use the lumped mass method, but this method fails to fully consider the role of the buoy and the contact between the cable and the seabed, resulting in the inability to accurately reflect the system behavior under actual working conditions. Existing models cannot accurately describe the non-linear behavior of the cable under seabed contact and embedding conditions, and cannot comprehensively reflect the dynamic characteristics of the buoy-mooring system. In the prior art, the influence of cable-seabed contact has also been considered, but most still focus on the contact between the cable and the seabed surface, ignoring the complex situation of partial cable embedding in the seabed. Therefore, how to accurately describe this complex behavior remains a difficult point in current cable modeling.

[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 tension characteristics of an additional buoy-type cable, which significantly improves the simulation accuracy.

[0005] Technical Solution: To achieve the above object, the present invention discloses a method for analyzing the tension characteristics of an additional buoy-type cable, including the following steps:

[0006] (1) Determine the parameters of the additional buoy-type cable model and the parameters of the sea conditions. The cable parameters include the coordinates of the upper end point of the cable, the coordinates of the lower end point of the cable, the cable length, the cable diameter, the unit length mass of the cable, the cable elastic modulus, the cable drag coefficient, the cable inertia coefficient, and the length of the part embedded in the seabed. The sea condition parameters include the water depth, the water density, and the water flow velocity;

[0007] (2) Set the parameters of the buoy according to the buoyancy-weight ratio. The parameters of the buoy include the installation distance of the buoy relative to the top of the cable, the mass of the buoy, and the volume of the buoy. The buoyancy-weight ratio is the ratio of the net buoyancy of the buoy to the net weight of the cable in water. The buoyancy-weight ratio is taken as 10% - 50%, and the net buoyancy of the buoy can be calculated. The mass and volume of the buoy are set based on the net buoyancy of the buoy. The ratio of the installation distance of the buoy relative to the top of the cable to the total length of the cable is equal to the buoyancy-weight ratio;

[0008] (3) Describe the movement of the cable element, and give the global position vector of any point P on the axis of the cable element in the initial configuration and the deformed configuration based on the absolute nodal coordinate method;

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

[0010] (5) Calculate the strain energy and elastic force of the cable element;

[0011] (6) Calculate the external load on the cable element to obtain the generalized external force matrix of the cable element. The external load includes the cable wet weight, additional buoy load, hydrodynamic resistance, and soil force;

[0012] (7) Assemble the mass matrix, absolute nodal coordinate vector, generalized external force matrix, and cable elastic force matrix of each element on the cable;

[0013] (8) Establish the cable dynamics equation of the additional buoy;

[0014] (9) Perform dynamic solution on the matrix form of the motion equations to obtain the absolute nodal coordinates of the cable element of the additional buoy, the position vector of the cable element during the simulation process, the cable elastic force, and the generalized external force. The tension on the cable element is the resultant force of the elastic force and the generalized external force of the cable element.

[0015] Optionally, step (3) specifically includes the following steps:

[0016] The global position vector r of any point P on the axis of the cable element before and after deformation is:

[0017] r = S e q e

[0018] In the formula: q e represents the nodal coordinates of the cable element, and S e represents the shape function of the cable element;

[0019] The shape function of the cable element is composed of the interpolation function S i (x) and is expressed as:

[0020] S e = [S1(x)IS2(x)IS3(x)IS4(x)I]

[0021] where I is a third-order identity matrix, and S i (x) is a function of the material coordinate and is independent of time. Here, x is the coordinate of any point P on the axis of the cable when the cable is undeformed in the cable element coordinate system;

[0022] If ξ = x / L, where L is the element length of the cable when it is undeformed, the element functions are:

[0023] S1 = 1 - 3ξ 2 + 2ξ 3 ; S1 = 1 - 3ξ 2 + 2ξ 3 ; S3 = 3ξ 2 - 2ξ 3; S4 = l(ξ 3 - ξ 2 )

[0024] The coordinates q of the unit nodes e can be composed of the node positions and their derivative components with respect to the material coordinate x, and can be expressed as:

[0025]

[0026] where q1 and q2 respectively represent the absolute node coordinates of the two endpoints of the unit, and r i , r j and r x,i , r x,j respectively represent the position vectors and gradient vectors of the first and last nodes i and j of the unit;

[0027] Since the shape function is independent of time, the velocity vector and acceleration vector of any point on the cable can be expressed as:

[0028]

[0029] where respectively represent the first and second derivatives of the absolute node coordinates of the unit with respect to time.

[0030] Optionally, the step (4) specifically includes the following steps:

[0031] The kinetic energy T of the cable unit e is expressed as:

[0032]

[0033] where M e is the mass matrix of the cable unit and is expressed as:

[0034]

[0035] In the formula: ρ is the density of the cable unit, and A is the cross-sectional area of the cable unit.

[0036] Optionally, the step (5) specifically includes the following steps:

[0037] The strain energy of the cable unit can be expressed by axial strain and bending curvature:

[0038]

[0039] Considering that the elastic force Q of the cable unit e is caused by unit bending and axial deformation, it can be calculated by taking the partial derivative of the strain energy with respect to the generalized coordinates:

[0040]

[0041] In the formula: A is the cross-sectional area of the cable, J is the polar moment of inertia of the unit cross-section, E is the elastic modulus of the cable, ε is the axial strain in the unit, and k represents the bending of the axis of the unit. Its expression is as follows:

[0042]

[0043] Among them, r′ and r″ represent the first and second partial derivatives of the position vector of any point P with respect to the arc length coordinate, expressed as:

[0044]

[0045] Among them, S′ and S″ are obtained by taking the first and second partial derivatives of the shape function S with respect to x.

[0046] Optionally, the wet weight W of the cable unit in step (6) can be the resultant force of the cable gravity f g and the buoyancy f b , as shown in the following formula:

[0047]

[0048] In the formula: g is the acceleration due to gravity, ρ r , ρ s are the cable density and seawater density respectively, and A is the outer circular cross-sectional area of the cable.

[0049] Optionally, in step (6), the buoy is directly attached to the corresponding unit of the cable in the form of a concentrated force, and the buoyancy on the unit where the buoy is located is converted into the corresponding additional buoy load Q qf , which can be expressed as:

[0050]

[0051] Among them, represents the shape function at the position of the node where the buoy is located, ρ s is the seawater density, g is the acceleration due to gravity, and V s is the volume of the buoy.

[0052] Optionally, in step (6), the cable is subject to the resistance of the fluid in water, which can be solved according to the Morison equation. Let the axial unit vector of the cable be i,

[0053] Then:

[0054]

[0055] Let:

[0056]

[0057] Then:

[0058] i = k(x)·r′

[0059] Then for any vector, its axial component is a·i, its axial projection is (a·i)·i, and its vertical projection is:

[0060] a - (a·i)·i = P·a

[0061] (P) k = P, P T = P

[0062] The normal and tangential ocean current forces on the cable unit can be expressed as:

[0063]

[0064] Where: U is the oncoming flow velocity, is the cable velocity; C f is the axial friction coefficient, C d is the radial drag coefficient, C m is the added mass coefficient.

[0065] Optionally, in step (6), a linear spring model is used to simulate the normal support force of the seabed on the cable. For any point P on the cable unit in contact with the seabed, the normal support force it receives is:

[0066] f z = k·z (z ≥ 0)

[0067] f z = 0 (z < 0)

[0068]

[0069] In the formula: k refers to the equivalent spring stiffness, d is the cable diameter, Z b is the seabed depth coordinate, e z is the unit direction vector in the Z direction;

[0070] When the cable is in the soil, it is only subject to the soil force. The soil resistance F soil received by the cable can be decomposed into two mutually perpendicular components, the normal resistance Q τ and the tangential resistance F l ;

[0071] Q τ = (E n D)qdl

[0072] F l = (E τD) fdl

[0073] where: E τ and E n are the effective width multipliers in the tangential and normal directions respectively, and dl represents the cable unit length;

[0074] The average normal pressure q and friction force f per unit area of the cable are:

[0075] f = S u

[0076] q = N c S u

[0077] where: S u is the undrained shear strength, and Nc is the bearing capacity factor;

[0078] Among them, the undrained shear strength of the soil is:

[0079] S u = S u0 - kz

[0080] where: S u0 is the undrained shear strength of the seabed surface, k is obtained from in-situ measurement, z is the depth, and z is taken as negative below the mud line;

[0081] The soil force acting on the cable in three-dimensional dynamic conditions is:

[0082]

[0083] Among them: τ, i are the unit vectors in the tangential and normal directions respectively. If n1 = [x1, y1, z1] represents the coordinates of node 1, then:

[0084]

[0085] Among them, A1 = E τ DN c S u0 dl; B1 = -E τ DN c kdl; A2 = -E i DS u0 dl; B2 = E i Dkdl

[0086] The undrained shear strength of the soil is determined by S u

[0087] Optionally, in step (7), it is assumed that the cable has n units, and the mass matrix of each module i is expressed as:

[0088] ​

[0089] Its overall mass matrix can be expressed as:

[0090]

[0091] The absolute nodal coordinate vector q and the generalized external force matrix Q of a single cable f And the assembly method of the cable elastic force matrix Q is the same as that of the mass matrix.

[0092] Optionally, in step (8), according to the principle of virtual work, the virtual work done by the cable is:

[0093]

[0094] Establish the dynamic equation of the cable according to the first type of Lagrange equation with constraint multipliers:

[0095]

[0096] Express the motion equation as a dynamic equation in matrix form:

[0097]

[0098] In the formula: M represents the cable mass matrix, Q represents the cable elastic force matrix, Q f Represents the generalized external force matrix, and the generalized external forces include the wet weight W, the additional buoy load Q qf , the hydrodynamic resistance f D And the soil force F soil ; C(q,t) represents the constraint equation, and C q Is the Jacobian matrix of the constraint equation; λ is the Lagrange multiplier related to the constraint equation; R is the second-order partial derivative of the constraint equation with respect to the independent variable, and its expression:

[0099]

[0100] Among them, Υ represents the opposite of the sum of all terms without the generalized acceleration term when taking the second-order derivative of the constraint equation with respect to time, C is the constraint equation, Is the first-order derivative of the constraint equation with respect to time, and α and β are correction factors, which are taken between 5 and 50, and the stable response is fast when α = β.

[0101] Advantages: Compared with the prior art, the present invention has the following remarkable advantages: Based on the ANCF method, the present invention accurately analyzes the dynamic behavior of the cable in a complex marine environment, can accurately describe the role of the buoy in the cable system, avoids the neglect of the role of the buoy in the traditional model, and thus significantly improves the simulation accuracy; The ANCF method adopted by the present invention can effectively capture the geometric deformation and dynamic response of the cable, is particularly suitable for large-scale dynamic simulation analysis, and details the contact situation between the cable and the seabed, making the present invention have high accuracy and reliability in dealing with and analyzing the non-linear behavior of the contact between the cable and the seabed; By analyzing different types of buoys and the installation positions of the buoys, the present invention reveals the influence of additional buoys on the cable tension characteristics. Brief Description of the Drawings

[0102] Figure 1 Schematic diagram of the mooring system with a buoy provided by the present invention;

[0103] Figure 2 3D mooring unit model diagram provided by the present invention;

[0104] Figure 3 Schematic diagram of the definition of external forces and tensions on the cable nodes provided by the present invention;

[0105] Figure 4 Schematic diagram of the seabed support force provided by the present invention;

[0106] Figure 5 Schematic diagram of the static configuration of the cable provided by the present invention;

[0107] Figure 6 Schematic diagram of the cable tension in Airy waves provided by the present invention;

[0108] Figure 7 Schematic diagram of the displacement of the observation points on the cables of different units provided by the present invention;

[0109] Figure 8 Schematic diagram of the static configuration and inclination distribution of the cable under different buoy sizes provided by the present invention;

[0110] Figure 9 Schematic diagram of the static configuration and inclination distribution of the cable at different positions provided by the present invention;

[0111] Figure 10 Schematic diagram of the influence of the buoy floating weight ratio and installation position on the cable tension characteristics provided by the present invention. Detailed Embodiment

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

[0113] A method for analyzing the tension characteristics of an additional buoy type cable, comprising the following steps:

[0114] (1) As Figure 1 shown, determine the parameters of the additional buoy type cable model and the sea state parameters. The cable parameters include the coordinates of the upper endpoint of the cable, the coordinates of the lower endpoint of the cable, the cable length, the cable diameter, the mass per unit length of the cable, the elastic modulus of the cable, the drag coefficient of the cable, the inertial coefficient of the cable, and the length of the part embedded in the seabed. The bottom of the cable is fixed to the seabed by an anchor pile, and an additional buoy is attached at a certain distance from the top of the cable. The additional buoy type cable model is located in the X-O-Z plane. The upper end of the cable is connected to a floating box. During the modeling process, a harmonic excitation is applied along the X direction at the top of the cable to simulate the actual movement of the floating box. The cable is divided into 60 units and numbered in sequence, and the corresponding node numbers are from N0 to N60. The sea state parameters include water depth, water density, and water flow velocity. The ocean current direction is consistent with the positive direction of the X axis, and the flow velocity is 1 m / s. The total length of the cable is 180 m, of which 20 m is embedded in the seabed and fixed to the seabed by an anchor pile. The cable parameters and the sea state parameters are shown in Table 1.

[0115] Table 1

[0116]

[0117]

[0118] (2) The parameters of the buoy are set according to the buoyancy-weight ratio, where the buoyancy-weight ratio is the ratio of the net buoyancy of the buoy to the net weight of the cable in water. The net buoyancy of the buoy is the resultant force of the buoyancy and gravity of the buoy, and the net weight of the cable in water is the resultant force of the buoyancy and gravity of the cable. The buoyancy-weight ratio is taken as 10% - 50%, and the net buoyancy of the buoy can be calculated. The mass and volume of the buoy are set based on the net buoyancy of the buoy. The installation position of the buoy is determined by the buoyancy-weight ratio. The ratio of the installation distance of the buoy relative to the top of the cable to the total length of the cable is equal to the buoyancy-weight ratio. The specific setting parameters of the buoy are shown in Table 2, including information such as the buoy type, size, and their corresponding optimal installation positions under different buoyancy-weight ratios.

[0119] Table 2

[0120]

[0121] (3) Describe the movement of the cable unit, and give the initial configuration and the deformed configuration of the cable unit based on the absolute nodal coordinate method, and the global position vector of any point P on the axis of the cable unit;

[0122] As Figure 2 shown, the configuration of the cable unit in the initial state and the deformed shape, where the global position vector r of any point P on the axis of the cable unit before and after deformation is:

[0123] r = S e q e

[0124] In the formula: q e represents the cable element node coordinates, and S e represents the shape function of the cable element;

[0125] The shape function of the cable element consists of the interpolation function S i (x) and is expressed as:

[0126] S e = [S1(x) I S2(x) I S3(x) I S4(x) I]

[0127] where I is a third-order identity matrix, and S i (x) is a function of the material coordinate and is independent of time, where x is the coordinate of an arbitrary point P on the axis of the cable in the cable element coordinate system when the cable is undeformed;

[0128] If ξ = x / L, where L is the element length of the cable when it is undeformed, the element function is:

[0129] S1 = 1 - 3ξ 2 + 2ξ 3 ; S1 = 1 - 3ξ 2 + 2ξ 3 ; S3 = 3ξ 2 - 2ξ 3 ; S4 = l(ξ 3 - ξ 2 )

[0130] The cable element node coordinates q e can be composed of the node position and its derivative components with respect to the material coordinate x, and can be expressed as:

[0131]

[0132] where q1 and q2 respectively represent the absolute node coordinates of the two endpoints of the element, and r i 、r j and r x,i 、r x,j respectively represent the position vectors and gradient vectors of the first and last nodes i and j of the element;

[0133] Since the shape function is independent of time, the velocity vector and acceleration vector of any point on the cable can be expressed as:

[0134]

[0135] where, respectively represent the first and second derivatives of the unit absolute nodal coordinates with respect to time;

[0136] (4) Construct the expressions for the kinetic energy and mass matrix of the cable element;

[0137] The kinetic energy T of the cable element e The expression is:

[0138]

[0139] where M e is the mass matrix of the cable element, expressed as:

[0140]

[0141] In the formula: ρ is the density of the cable element, and A is the cross-sectional area of the cable element;

[0142] (5) Calculate the strain energy and elastic force of the cable element;

[0143] Since the cable element neglects the torsional deformation, the strain energy of the cable element can be expressed by the axial strain and the bending curvature:

[0144]

[0145] Considering the elastic force Q of the cable element e is caused by the bending and axial deformation of the element, and can be calculated by taking the partial derivative of the strain energy with respect to the generalized coordinates:

[0146]

[0147] In the formula: A is the cross-sectional area of the cable, J is the polar moment of inertia of the element cross-section, E is the elastic modulus of the cable, ε is the axial strain in the element, and k represents the bending of the axis of the element. Its expression is as follows:

[0148]

[0149] where r′ and r″ represent the first and second partial derivatives of the position vector of any point P with respect to the arc length coordinate, expressed as:

[0150]

[0151] where S′ and S″ are obtained by taking the first and second partial derivatives of the shape function S with respect to x;

[0152] (6) Calculate the external loads on the cable element to obtain the generalized external force matrix of the cable element. The external loads include the wet weight of the cable element, the additional buoy load, the hydrodynamic resistance, and the soil force; As Figure 3 shown, it is a schematic diagram of the external forces and tensions on the cable element, showing the force conditions of the underwater cable element and the components of the tension in each direction.

[0153] (6.1) The wet weight W of the cable unit can be obtained from the resultant force of the cable gravity f g and the buoyancy f b as follows:

[0154]

[0155] where: g is the acceleration due to gravity, ρ r , ρ s are the cable density and seawater density respectively, and A is the cross-sectional area of the outer circle of the cable;

[0156] (6.2) Additional buoy load Q qf :

[0157] The buoy is directly attached to the corresponding unit of the cable in the form of a concentrated force, and the buoyancy on the unit where the buoy is located is converted into the corresponding additional buoy load Q qf , which can be expressed as:

[0158]

[0159] where, represents the shape function at the node position where the buoy is located, ρ s is the seawater density, g is the acceleration due to gravity, and V s is the volume of the buoy;

[0160] (6.3) Hydrodynamic resistance f D :

[0161] Since the cable is a relatively small-scale component, the cable is subject to fluid resistance in water, and the calculation can be performed according to the Morison equation. Let the axial unit vector of the cable be i,

[0162] then:

[0163]

[0164] Let:

[0165]

[0166] then:

[0167] i = k(x)·r′

[0168] Then for any vector, its axial component is a·i, its axial projection is (a·i)·i, and its vertical projection is:

[0169] a - (a·i)·i = P·a

[0170] (P) k = P, P T = P

[0171] The normal and tangential ocean current forces on the cable unit can be expressed as:

[0172]

[0173] Where, U is the incoming flow velocity, is the cable velocity; C f is the axial friction coefficient, C d is the radial drag coefficient, C m is the added mass coefficient; the water flow resistance on the cable unit can be calculated based on the normal and tangential ocean current forces.

[0174] (6.4) Soil force F soil

[0175] As Figure 4 shown, when the cable is located on the seabed, the influence of soil friction resistance on the cable is ignored, and a linear spring model is used to simulate the normal supporting force of the seabed on the cable. For any point P on the cable unit in contact with the seabed, the normal supporting force it receives is:

[0176] f z = k·z (z≥0)

[0177] f z = 0 (z≥0)

[0178]

[0179] In the formula: k refers to the equivalent stiffness of the spring, d is the cable diameter, Z b is the seabed depth coordinate, e z is the unit direction vector in the Z direction.

[0180] When the cable is located in the soil, it is only affected by soil forces. The soil resistance F soil on the cable can be decomposed into two mutually perpendicular components, the normal resistance Q τ and the tangential resistance F l .

[0181] Q τ =(E n D)qdl

[0182] F l =(E τ D)fdl

[0183] In the formula: E τ and E m are the effective width multipliers in the tangential and normal directions respectively, and dl represents the cable unit length;

[0184] The average normal pressure q and frictional force f per unit area of the cable are:

[0185] f = S u

[0186] q = N c S u

[0187] Where: S u is the undrained shear strength, and Nc is the bearing capacity factor;

[0188] Among them, the undrained shear strength of the soil is:

[0189] S u = S u0 - kz

[0190] Where: S u0 is the undrained shear strength of the seabed surface, k is obtained from in-situ measurement, z is the depth, and z is taken as negative below the mud line;

[0191] The soil force on the cable in three-dimensional dynamic conditions is:

[0192]

[0193] Among them: τ and i are the unit vectors in the tangential and normal directions respectively. If n1 = [x1, y1, z1] represents the coordinates of node 1, then:

[0194]

[0195] Among them, A1 = E τ DN c S u0 dl; B1 = -E τ DN c kdl; A2 = -E i DS u0 dl; B2 = E i Dkdl

[0196] The undrained shear strength of the soil is determined by S u = -1.45z kPa, E i , E τ are taken as 8.0 and 2.5 respectively, and N c at any depth is taken as 7.6;

[0197] (7) Assemble the mass matrix, absolute nodal coordinate vector, generalized external force matrix, and cable elastic force matrix of each unit on the cable;

[0198] Assume that the cable has n units, and the mass matrix of each module i is expressed as:

[0199]

[0200] Its overall mass matrix can be expressed as:

[0201]

[0202] The absolute nodal coordinate vector q and the generalized external force matrix Q of a single cable f And the assembly method of the cable elastic force matrix Q is the same as that of the mass matrix.

[0203] (8) Establish the cable dynamics equation of the additional buoy;

[0204] According to the principle of virtual work, the virtual work done by the cable is:

[0205]

[0206] Establish the dynamics equation of the cable according to the first type of Lagrange equation with constraint multipliers:

[0207]

[0208] Express the motion equation as a matrix form of the dynamics equation:

[0209]

[0210] In the formula: M represents the cable mass matrix, Q represents the cable elastic force matrix, Q f represents the generalized external force matrix, and the generalized external forces include the wet weight W, the additional buoy load Q qf , the hydrodynamic resistance f D and the soil force F soil ; C(q,t) represents the constraint equation, C q is the Jacobian matrix of the constraint equation; λ is the Lagrange multiplier related to the constraint equation; R is the second-order partial derivative of the constraint equation with respect to the independent variable, and its expression:

[0211]

[0212] Among them, γ represents the opposite number after adding all the terms without the generalized acceleration term when taking the second-order derivative of the constraint equation with respect to time, C is the constraint equation, is the first-order derivative of the constraint equation with respect to time, α and β are correction factors, and they are taken between 5 and 50. When α = β, the stable response is relatively fast;

[0213] (9) Use the ode113 solver to perform dynamic solution on the matrix form of the motion equations to obtain the absolute nodal coordinates q of the additional buoy cable element e , through q eThe position vector r, the cable elastic force Q, and the generalized external force Q of the cable element during the simulation can be calculated. f Among them, the tension force on the cable element is the resultant force of the elastic force Q and the generalized external force Q of the cable element. f of the resultant force.

[0214] As Figure 5 shown, to verify the accuracy of the modeling method of the present invention, the figure shows that the underwater static configuration of a single cable line is consistent between the ANCF theoretical model and the OrcaFlex software simulation results, and the static characteristics of the cable can be accurately analyzed, and the consistency and accuracy of the two methods in the calculation of the static configuration can be verified.

[0215] As Figure 6 shown, by adding buoy III at a distance of 29.2 meters from the upper end of the cable and performing regular wave simulation with Airy waves with an amplitude of 5 meters and a period of 12.5 seconds along the positive X-axis, it is found that the ANCF theoretical model and the OrcaFlex software simulation results are consistent.

[0216] As Figure 7 shown, by applying a harmonic motion X = A f sin(2πft) along the X direction at the top of the cable, with an amplitude A f = 3m and a frequency f = 0.08Hz; selecting node N48 as the observation point and comparing the convergence of the displacement in the X direction of the observation point in the ANCF model and the OrcaFlex software; the results show that the present invention can maintain high precision even under a low number of element divisions, significantly superior to the convergence performance of the OrcaFlex software.

[0217] As Figure 8 shown, attaching buoys with different buoyant weight ratios can significantly change the contact length between the cable and the seabed and its static tension distribution; when the buoyant weight ratio is less than 40%, the number of contact segments between the cable and the seabed increases with the increase of the buoyant weight ratio, and the static tension of each segment can be reduced by up to 30%.

[0218] As Figure 9 shown, the installation position of the buoy is crucial for the static performance of the cable. When the buoy is installed closer to the upper end point of the cable, the number of segments in contact with the seabed is more; when the installation position of the buoy is more than 72 meters away from the upper end point, this effect weakens and the number of contact segments decreases accordingly.

[0219] As Figure 10 shown, the buoyant weight ratio and the installation position jointly determine the tension characteristics of the cable. Installing the buoy at position III can most effectively reduce the static tension of the whole section, while when at position II, the tension distribution of the suspended section is the most uniform; the results show that by precisely adjusting the buoyant weight ratio and the installation position, not only can the tension be significantly reduced, but also the dynamic response characteristics of the mooring system can be improved.

Claims

1. A method for analyzing the tension characteristics of an additional buoy type cable, characterized in that, It includes the following steps: (1) Determine the parameters of the additional buoyant cable model and the sea state parameters, where the cable parameters include the coordinates of the upper endpoint of the cable, the coordinates of the lower endpoint of the cable, the cable length, the cable diameter, the unit length quantity of the cable, the elastic modulus of the cable, the cable drag coefficient, the cable inertia coefficient, and the length of the part embedded in the seabed, and the sea state parameters include water depth, water density, and water flow velocity; (2) The parameters of the buoy are set according to the buoyancy-weight ratio. The parameters of the buoy include the installation distance of the buoy relative to the top of the cable, the mass of the buoy, and the volume of the buoy; where the buoyancy-weight ratio is the ratio of the net buoyancy of the buoy to the net weight of the cable in water. The buoyancy-weight ratio is taken as 10% - 50%, and the net buoyancy of the buoy can be calculated. The mass and volume of the buoy are set based on the net buoyancy of the buoy. The ratio of the installation distance of the buoy relative to the top of the cable to the total length of the cable is equal to the buoyancy-weight ratio; (3) Describe the motion of the cable element, and based on the absolute nodal coordinate method, give the global position vector of any point P on the axis of the cable element in the initial configuration and the deformed configuration; (4) Construct the expressions for the kinetic energy and mass matrix of the cable element; (5) Calculate the strain energy and elastic force of the cable element; (6) Calculate the external loads acting on the cable element to obtain the generalized external force matrix of the cable element. The external loads include the wet weight of the cable, the additional buoy load, hydrodynamic resistance, and soil force; (7) Assemble the mass matrix, absolute nodal coordinate vector, generalized external force matrix, and cable elastic force matrix of each element on the cable; (8) Establish the cable dynamics equation of the additional buoy; (9) Perform dynamic solution on the system of motion equations in matrix form to obtain the absolute nodal coordinates of the additional buoy cable element, the position vector of the cable element during the simulation process, the cable elastic force, and the generalized external force, where the tension acting on the cable element is the resultant force of the elastic force and the generalized external force of the cable element.

2. The method for analyzing the tension characteristics of a cable with an additional buoy according to claim 1, wherein The specific steps of step (3) include the following steps: The global position vector r of any point P on the axis of the cable element before and after deformation is: r = S e q e Where: q e represents the node coordinates of the cable element, and S e represents the shape function of the cable element; The shape function of the cable unit consists of the interpolation function S i (x) and is expressed as: S e = [S1(x) IS2(x) IS3(x) IS4(x) I] where I is a third-order identity matrix, and S i (x) is a function of the material coordinates and independent of time, where x is the coordinate of any point P on the central axis of the cable when it is undeformed in the cable element coordinate system; If ξ = x / L, where L is the length of the cable element before deformation, the element function is: S1 = 1 - 3ξ 2 + 2ξ 3 ; S1 = 1 - 3ξ 2 + 2ξ 3 ; S3 = 3ξ 2 - 2ξ 3 ; S4 = l(ξ 3 - ξ 2 ) Unit node coordinate q e It can be composed of the node position and its derivative components with respect to the material coordinate x, and can be expressed as: where q1 and q2 respectively represent the absolute nodal coordinates of the two endpoints of the element, and r i , r j and r x,i , r x,j respectively represent the position vectors and gradient vectors of the first and last nodes i and j of the element; Since the shape function is independent of time, the velocity vector and acceleration vector at any point on the cable can be expressed as: Among them, respectively represent the first and second order derivatives of the unit absolute nodal coordinates with respect to time.

3. The method for analyzing the tension characteristics of an additional buoy type cable according to claim 2, wherein The specific steps of step (4) include the following steps: Cable unit kinetic energy T e The expression is: Among them, M e is the mass matrix of the cable unit, expressed as: In the formula: ρ is the density of the cable element, and A is the cross-sectional area of the cable element.

4. An additional buoy type cable tension characteristic analysis method according to claim 3, characterized in that The specific steps of step (5) include the following steps: The strain energy of the cable element can be expressed by the axial strain and bending curvature: Consider the elastic force Q of the cable element e which is caused by the bending and axial deformation of the element and can be calculated by taking the partial derivative of the strain energy with respect to the generalized coordinates: In the formula: A is the cross-sectional area of the cable, J is the polar moment of inertia of the unit cross-section, E is the elastic modulus of the cable, ε is the axial strain in the element, k represents the bending of the axis of the element, and its expression is as follows: Among them, r′ and r″ represent the first and second partial derivatives of the position vector of any point P with respect to the arc length coordinate, expressed as: Among them, S′ and S″ are obtained by taking the first and second partial derivatives of the shape function S with respect to x.

5. The method for analyzing the tension characteristics of an additional buoy type cable according to claim 4, wherein In the step (6), the wet weight W of the cable unit can be the resultant force of the cable gravity f g and the buoyancy f b , as shown in the following formula: where: g is the acceleration due to gravity, ρ r , ρ s are the density of the cable and the density of seawater respectively, and A is the cross-sectional area of the outer circle of the cable.

6. The method for analyzing the tension characteristics of an additional buoy type cable according to claim 5, wherein, In step (6), the buoy is directly attached to the corresponding unit of the cable in the form of a concentrated force, and the buoyancy on the unit where the buoy is located is converted into the corresponding additional buoy load It can be expressed as: Among them, is the shape function indicating the position of the buoy at the node, ρ s is the seawater density, g is the acceleration due to gravity, V s is the volume of the buoy.

7. A method for analyzing the tension characteristics of an additional buoy type cable according to claim 6, characterized in that, In step (6), the cable is subject to the resistance of the fluid in water, which can be solved and calculated according to the Morison equation. Let the axial unit vector of the cable be i, Then: Let: Then: i = k(x)·r′ Then for any vector, its axial component is a·i, its axial projection is (a·i)·i, and its vertical projection is: a - (a·i)·i = P·a (P) k = P, P T = P The normal and tangential ocean current forces acting on the cable element can be expressed as: Among them, U is the oncoming flow velocity, is the cable velocity; C f is the axial friction coefficient, C d radial drag force coefficient, C m added mass coefficient.

8. An additional buoy type cable tension characteristic analysis method according to claim 7, characterized in that In step (6), a linear spring model is used to simulate the normal support force of the seabed on the cable. For any point P on the cable element in contact with the seabed, the normal support force it receives is as follows: f z = k·z (z ≥ 0) f z = 0 (z ≥ 0) Δz = r p ·e z -Z b , Where: k refers to the equivalent stiffness of the spring, d is the cable diameter, Z b is the seabed depth coordinate, e z is the unit direction vector in the Z direction; When the cable is located in the soil, it is only subjected to the action of the soil, and the soil resistance F soil acting on the cable, according to the research of Vivatrat et al., can be decomposed into two mutually perpendicular components, the normal resistance Q τ and the tangential resistance F l ; Q τ = (E n D)qdl F l = (E τ D) fdl where: E τ and E n are the effective width multipliers in the tangential and normal directions respectively, and dl represents the cable unit length; The average normal pressure q and friction force f per unit area of the cable are: f = S u q = N c S u Where: S u is the undrained shear strength, and Nc is the bearing capacity factor; Among them, the undrained shear strength of the soil is: S u = S u0 -kz Where: S u0 is the undrained shear strength of the seabed surface, k is obtained from in-situ measurement, z is the depth, and z is taken as negative below the mud line; The soil force received by the cable under three-dimensional dynamic conditions is: Among them: τ and i are the tangential and normal unit vectors respectively. If n1 = [x1, y1, z1] represents the coordinates of node 1, then: where, A1 = E τ DN c S u0 dl; B1 = -E τ DN c kdl; A2 = -E i DS u0 dl; B2 = E i DkDl The undrained shear strength of the soil is determined by S u to be determined 9. An additional buoy type cable tension characteristic analysis method according to claim 8, characterized in that, In step (7), it is assumed that the cable has n elements, and the mass matrix of each module i is expressed as: Its overall mass matrix can be expressed as: The absolute nodal coordinate vector q and the generalized external force matrix Q of a single cable f And the assembly method of the cable elastic force matrix Q is the same as that of the mass matrix.

10. A method for analyzing the tension characteristics of an additional buoy-type cable according to claim 9, characterized in that, In step (8), according to the principle of virtual work, the virtual work done by the cable is: Establish the dynamic equation of the cable according to the first type of Lagrange equation with constraint multipliers: Express the motion equation as a dynamic equation in matrix form: where: M represents the cable mass matrix, Q represents the cable elastic force matrix, Q f represents the generalized external force matrix, and the generalized external forces include the wet weight W, the additional buoy load the hydrodynamic drag f D and the soil force F soil ; C(q,t) represents the constraint equation, and C q is the Jacobian matrix of the constraint equation; λ is the Lagrange multiplier related to the constraint equation; R is the second-order partial derivative of the constraint equation with respect to the independent variable, and its expression is: Among them, γ represents the opposite number of the sum of all terms without generalized acceleration terms when the constraint equation is differentiated twice with respect to time, C is the constraint equation, is the first derivative of the constraint equation with respect to time, α and β are correction factors, which are taken between 5 and 50, and the stable response is fast when α = β.

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