Method and system for modeling dynamics of slender cable based on global lumped mass method

Abstract

This invention discloses a method and system for dynamic modeling of slender cables based on the global lumped mass method. The method includes: dividing the cable into multiple cable elements using the lumped mass method in a global coordinate system; obtaining the node coordinates of the cable elements and their relative velocity vectors in the fluid medium; using the global cable node coordinates, obtaining the tangential vector and normal plane of the cable elements; projecting the relative velocity vectors into their tangential and normal components; and using the right-hand rule to obtain the relative velocity tangential vector, normal vector, and binormal vector in the global coordinate system; calculating the dynamic loads and vortex-induced vibrations of the cable elements; and calculating the tension, damping force, buoyancy, and gravity loads of the cable elements to construct a cable dynamic model in the fluid medium; finally, constructing a variable-length cable model using the arbitrary Lagrange-Euler method. In this invention, all loads are expressed in a global coordinate system, eliminating the need for coordinate transformation, avoiding singularities, and resulting in higher computational efficiency.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling technology for slender cables, and in particular to a method and system for dynamic modeling of slender cables based on the global lumped mass method. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] With the development of marine engineering and marine exploration technologies, slender cable structures have found increasing applications, such as mooring cables for floating platforms and marine towing cables. Similarly, with technological advancements, slender cable structures are also widely used in other fields, such as aerial refueling in the aerospace field.

[0004] Dynamic models of slender cables in fluid media (including liquid and gaseous fluids) are crucial for obtaining the dynamic response of equipment systems. Currently, the lumped mass method is a commonly used approach for constructing dynamic models of cables in fluid media. The traditional lumped mass method divides the cable into a finite number of elements, constructing element coordinates such as Euler angular coordinates, Frenet's method (i.e., polar coordinates), and relative velocity element coordinates. These element coordinates are used to express the cable morphology and internal and external forces (including tension, damping force, gravity, buoyancy, and hydrodynamic loads). Euler angle coordinates describe the element's attitude using two Euler angles formed by the nodal position coordinates. While this method makes the element's attitude relatively easy to understand, singularities can occur under certain special attitudes due to the inverse trigonometric function calculations. The Frenet method uses coordinates based on the shape of the entire cable to describe the element's attitude. Its orthogonal basis vectors are the element's tangent, normal, and binormal directions. The normal direction can be obtained from the tangent vector, and the binormal direction can be defined by the vector product of the tangent and normal directions. However, when the curvature of the cable's shape function is zero, meaning some or all nodes are on a straight line, the normal vector direction cannot be determined, and the Frenet method also exhibits singularities. The relative velocity element method uses coordinates based on a plane composed of the element's tangent and relative velocity directions to describe the element's attitude. However, if the tangent and relative velocity directions are parallel, this plane cannot be constructed. Therefore, the three commonly used coordinate systems mentioned above are singular under certain special attitudes, and these singularities can make solving the cable's dynamic equations difficult. Therefore, there is an urgent need for a singularity-free method to construct cable dynamics models, so as to more robustly and efficiently express the elastic deformation load of the cable itself and the load of the surrounding fluid medium, and facilitate the solution of cable dynamic characteristics.

[0005] Furthermore, depending on the operational requirements, it is often necessary to change the length of cables in fluid media, such as the cable retrieval and deployment operations of marine towing systems, the retrieval and deployment operations of tower cranes and aerial refueling hoses, and the laying operations of cables and pipelines. Therefore, constructing a dynamic model for cables of variable length and mass is crucial for simulating cable retrieval and deployment operations.

[0006] Furthermore, when there is a relative velocity between the slender cable and the surrounding medium, vortex shedding often occurs, generating periodically varying lift in the direction perpendicular to the relative velocity, which in turn causes vortex-induced vibration of the cable in that direction. Since the cable moves freely and deforms in three-dimensional space, the relative velocity direction between the cable and the fluid medium is uncertain, requiring the use of a unit coordinate system to represent fluid resistance and lift. However, because the existing unit coordinate system is independent of relative velocity, complex unit coordinate transformations are needed to obtain the load applied in the direction perpendicular to the relative velocity. Therefore, how to efficiently and accurately represent the load in the direction perpendicular to the relative velocity is key to studying the vortex-induced vibration of slender cables. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention provides a method and system for modeling the dynamics of slender cables based on the global lumped mass method. It improves upon the relative velocity coordinate system to obtain a global lumped mass method in a global coordinate system. This method is then applied to cable dynamics modeling in fluid media, providing a more robust and efficient representation of the morphology, internal forces, and external forces of slender cables within the fluid. Compared to traditional coordinate systems, this global lumped mass method avoids singularities and improves computational efficiency due to minimal coordinate changes. Furthermore, combining the arbitrary Lagrange-Eulerian method with the global lumped mass method (ALE-GLMF) for modeling variable-length cables ensures model accuracy while improving computational efficiency and reducing computation time. Based on the global coordinate system, the vortex-induced vibration of slender cables can be represented.

[0008] In a first aspect, the present invention provides a method for dynamic modeling of slender cables based on the global lumped mass method.

[0009] A method for modeling the dynamics of slender cables based on the global lumped mass method includes:

[0010] In the global coordinate system, the shape of the flexible slender cable in the fluid medium is obtained. The cable is divided into multiple cable units by the lumped mass method. The position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium are obtained.

[0011] Based on the position coordinates of the nodes at both ends of the cable unit, the tangential vector and normal plane of the cable unit are obtained. The relative velocity vector is projected into the tangential component and normal component of the relative velocity using the tangential vector and normal plane. Then, the unit tangential vector and unit normal vector of the relative velocity are determined. The unit binormal vector of the relative velocity is obtained according to the right-hand rule. In this way, the fluid resistance, lift and additional mass load acting on the cable unit are calculated.

[0012] Based on the position coordinates of the nodes at both ends of the cable unit, the tension, damping force, buoyancy and gravity loads acting on the cable unit are obtained;

[0013] Based on the calculation results of all cable elements, the results are substituted into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0014] A further technical solution involves, based on the calculated loads of all cable units and combined with the mass of all cable units, distributing the load and mass of each cable unit equally to the nodes at both ends of the cable unit, and substituting the loads and masses acting on the nodes into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0015] Secondly, the present invention provides a dynamic modeling system for slender cables based on the global lumped mass method.

[0016] A dynamic modeling system for slender cables based on the global lumped mass method includes:

[0017] The global position coordinate acquisition module is used to acquire the shape of the flexible slender cable in the fluid medium in the global coordinate system. The cable is divided into multiple cable units by using the lumped mass method. The position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium are acquired.

[0018] The calculation module is used to obtain the tangential vector and normal plane of the cable unit based on the position coordinates of the nodes at both ends of the cable unit. It then projects the relative velocity vector into the tangential and normal components of the relative velocity using the tangential vector and normal plane, thereby determining the unit tangential vector and unit normal vector of the relative velocity. Finally, it obtains the unit binormal vector of the relative velocity according to the right-hand rule, and uses this to calculate the fluid resistance, lift, and additional mass load acting on the cable unit. Based on the position coordinates of the nodes at both ends of the cable unit, it obtains the tension, damping force, buoyancy, and gravity load acting on the cable unit.

[0019] The modeling module is used to substitute the calculation results of all cable elements into the motion equation of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0020] A further technical solution involves, based on the calculated loads of all cable units and combined with the mass of all cable units, distributing the load and mass of each cable unit equally to the nodes at both ends of the cable unit, and substituting the loads and masses acting on the nodes into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0021] Thirdly, this disclosure also provides an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the steps of the method described in the first aspect.

[0022] Fourthly, this disclosure also provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the steps of the method described in the first aspect.

[0023] The above one or more technical solutions have the following beneficial effects:

[0024] 1. This invention provides a method and system for dynamic modeling of slender cables based on the global lumped mass method. By improving the existing relative velocity element coordinate system, a global lumped mass method based on a global coordinate system is obtained. This method uses global node coordinates to calculate the tension, damping force, buoyancy, gravity, and other loads of the cable element. Compared to the traditional coordinate system, the global lumped mass method does not employ coordinate transformation, fundamentally eliminating the singularity of the traditional element coordinate system and further improving computational efficiency.

[0025] 2. This invention utilizes global node coordinates to represent the tangential vector and normal plane of the cable element, and projects the relative velocity into tangential and normal components using these coordinates. This yields the relative velocity tangential and normal vectors based on global coordinates, thereby expressing the cable's tangential fluid resistance, normal fluid resistance, and additional mass load. Compared to traditional coordinate systems, the global lumped mass method does not employ coordinate transformation, improving the computational efficiency of hydrodynamic loads and eliminating singularities. Furthermore, based on the right-hand rule, this invention uses the relative velocity tangential and normal vectors in global coordinates to obtain a secondary normal vector. This secondary normal vector is perpendicular to the relative velocity direction, facilitating the expression of the lift force causing vortex-induced vibration. Compared to traditional coordinate systems, the global lumped mass method does not employ coordinate transformation, improving the computational efficiency of lift and vortex-induced vibration and eliminating singularities.

[0026] 3. Based on the global lumped mass method, this invention combines the arbitrary Lagrange-Euler method to propose the Arbitrary Lagrange-Euler Global Lumped Mass Method (ALE-GLMF). This method is used to model cables of varying lengths in fluid media. During the modeling process, each node of the cable element has only 4 coordinates (including 3 position coordinates and 1 material coordinate). This method is used to model cables with small bending moduli in fluid media, which can effectively improve computational efficiency and reduce computation time while ensuring modeling accuracy. Attached Figure Description

[0027] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0028] Figure 1 This is a flowchart of the slender cable dynamics modeling method based on the global lumped mass method described in an embodiment of the present invention;

[0029] Figure 2 This is a schematic diagram of the spatial discretization of the lumped mass method in an embodiment of the present invention;

[0030] Figure 3 This is a schematic diagram showing the decomposition of relative velocity, fluid resistance, and secondary normal lift under the global lumped mass method in an embodiment of the present invention;

[0031] Figure 4 This is a schematic diagram of the local and global coordinate systems of the Euler method in an embodiment of the present invention;

[0032] Figure 5 This is a schematic diagram of the Frenet coordinate system in an embodiment of the present invention;

[0033] Figure 6 This is a schematic diagram of the orthogonal basis of the relative velocity unit coordinate system in an embodiment of the present invention;

[0034] Figure 7 This is a schematic diagram of the simulation results of the cable dynamics model in an embodiment of the present invention;

[0035] Figure 8 This is a schematic diagram illustrating the singularity range of the Euler coordinate system in an embodiment of the present invention;

[0036] Figure 9 This is a schematic diagram of the position vector of a node on cable element i under any Lagrange-Eulerian global lumped mass method in an embodiment of the present invention;

[0037] Figure 10 This is a schematic diagram of the simulation results of the dynamic model of the variable length cable in an embodiment of the present invention. Detailed Implementation

[0038] It should be noted that the following detailed descriptions are exemplary and are intended only to describe specific embodiments and to provide further explanation of the invention, and are not intended to limit the scope of exemplary embodiments of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0039] The lumped mass method is widely used for dynamic modeling of slender cables in fluid media (including underwater and air). To efficiently represent cable element loads, coordinate systems such as the Euler coordinate system, the Frenet coordinate system, and the relative velocity element coordinate system have been proposed. However, almost all element coordinate systems exhibit singularities when representing the transformation matrix of values ​​transferred between the element and the global coordinate system. Therefore, this invention improves the relative velocity element coordinate system, obtaining a global lumped mass method based on the global coordinate system. This global lumped mass method more robustly and efficiently represents the dynamic loads related to the relative velocity and acceleration of the cable in the flow field. Specifically, in the global lumped mass method, the tangential vector and normal plane of the cable element are represented using global node coordinates. The relative velocity is then projected into tangential and normal components using the tangential vector and normal plane, thereby obtaining the relative velocity tangential vector and normal vector per unit length. The right-hand rule is then used to obtain the secondary normal vector, which is used to calculate the normal fluid resistance, tangential fluid resistance, lift, and additional mass loads related to fluid acceleration acting on the cable element. At the same time, the tension, damping force, buoyancy, and gravity of the cable element are calculated using global node coordinates in the global coordinate system. Compared with the traditional coordinate system, the global lumped mass method does not use coordinate transformation, thus fundamentally eliminating the singularity of the traditional element coordinate system and further improving computational efficiency.

[0040] Furthermore, existing modeling methods for variable-length flexible cables suffer from drawbacks such as the high computational cost and the unsatisfactory determination of the cable's dynamic response. This invention, based on the aforementioned global lumped mass method and incorporating the arbitrary Lagrange-Eulerian method, proposes a new coordinate system. In this system, each node of the cable element has only four coordinates (three positional coordinates and one material coordinate). This system effectively improves computational efficiency and reduces computation time while ensuring modeling accuracy.

[0041] Example 1

[0042] This embodiment provides a method for modeling the dynamics of slender cables based on the global lumped mass method, such as... Figure 1 As shown, the specific steps include:

[0043] Step S1: In the global coordinate system, obtain the shape of the flexible slender cable in the fluid medium. Use the lumped mass method to divide the cable into multiple cable units. Obtain the position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium.

[0044] Step S2: Based on the position coordinates of the nodes at both ends of the cable unit, obtain the tangential vector and normal plane of the cable unit. Use the tangential vector and normal plane to project the relative velocity vector into the tangential component and normal component of the relative velocity, and then determine the unit tangential vector and unit normal vector of the relative velocity. According to the right-hand rule, obtain the unit binormal vector of the relative velocity, and use this to calculate the fluid resistance, lift and additional mass load acting on the cable unit.

[0045] Step S3: Based on the position coordinates of the nodes at both ends of the cable unit, obtain the tension, damping force, buoyancy and gravity loads acting on the cable unit;

[0046] Step S4: Substitute the calculation results of all cable units into the motion equation of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0047] In a further technical solution, in step S4 above, based on the calculated loads of all cable units and combined with the mass of all cable units, the load and mass of each cable unit are evenly distributed to the nodes at both ends of the cable unit. The loads and masses acting on the nodes are then substituted into the motion equation of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0048] Combining the application of slender cable structures in fields such as marine engineering and marine exploration, and taking the dynamic modeling of slender cables in liquid fluids as an example, the following content will provide a more detailed introduction to the dynamic modeling method of slender cables based on the global lumped mass method proposed in this embodiment.

[0049] In step S1, the morphology of the flexible, slender cable in the fluid medium is obtained in the global coordinate system. The cable is divided into multiple cable units using the lumped mass method, such as... Figure 2 As shown. The unit forces distributed at the nodes at both ends of each cable unit include tension, damping force, gravity, buoyancy and hydrodynamic force. Among them, the forces acting on the unit can be divided into internal forces and external forces. The internal forces are tension and damping force, and the external forces are gravity, buoyancy and hydrodynamic force.

[0050] Obtain the position coordinates of the two end nodes of each cable element in the global coordinate system, as well as the relative velocity vector of the cable element in the fluid medium. Specifically, such as... Figure 3As shown, the XYZ coordinate system represents the global coordinate system. For the i-th cable element, the nodes at both ends of the cable element are node i and node i+1, respectively. The position vectors of the cable element i at both ends of the cable element in the global coordinate system are obtained, where the position vector of node i is r. i The position vector of node i+1 is r i+1 ; Obtain the relative velocity vector v of cable element i in the fluid medium in the global coordinate system. g .

[0051] In steps S2 and S3, calculations are performed based on the three obtained position coordinates. Specifically, based on the position coordinates of the nodes at both ends of the cable unit, the tangential vector and normal plane of the cable unit are obtained. The relative velocity vector is projected into the tangential component and normal component of the relative velocity using the tangential vector and normal plane, thereby determining the unit tangential vector and unit normal vector of the relative velocity. The unit binormal vector of the relative velocity is obtained using the right-hand rule, thereby obtaining the hydrodynamic loads acting on the cable unit, including fluid resistance (including normal fluid resistance and tangential fluid resistance), lift (i.e., vertical lift), and additional mass load (i.e., normal additional mass load). Based on the position coordinates of the nodes at both ends of the cable unit, the tension, damping force, buoyancy, and gravity loads acting on the cable unit are calculated.

[0052] In the global lumped quality method proposed in this embodiment, such as Figure 3 As shown, the force acting on the cable element and the element mass matrix are represented by the node position. As shown in equation (1), firstly, based on the position coordinates r of node i... i and the position coordinates r of node i+1 i+1 Determine the direction vector E of the cable element from node i to node i+1. g Calculate the unit direction vector z based on this direction vector. g , as the tangential vector of the cable element; calculate the normal plane based on the tangential vector (the projection matrix of the normal plane is P). g Then, using the tangential vector z g The projection matrix P of the normal plane g The relative velocity vector v g The projection is the tangential component of the relative velocity v gt and the relative velocity normal component v gn The fluid resistance and additional mass inertial force acting on the cable unit are calculated based on the tangential and normal components of the relative velocity, thus obtaining the mass matrix M of the cable unit. g Tangential water resistance F acting on the cable unit Dt Normal water resistance F Dn Vertical lift F L The normal additional mass inertial force F caused by fluid accelerationag The specific calculation formula is as follows:

[0053]

[0054] In the above formula, P g Let v be the projection matrix onto the normal plane. w Let a be the velocity of the fluid medium in global coordinates. w C represents the acceleration of the fluid medium in global coordinates. L C is the lift coefficient. f and C n These are the tangential drag coefficient and the normal drag coefficient, respectively. m It is the additional mass coefficient, ρ c and ρ f For the density of the cable and the fluid medium, I 3×3 It is a third-order identity matrix. d is the velocity of cable node i in global coordinates. c is the cable diameter, and l represents the current length of the cable unit.

[0055] Secondly, based on the position coordinates of the nodes at both ends of the cable unit, the tension, damping force, buoyancy, and gravity loads acting on the cable unit are calculated. This is based on the tangential vector z. g The tension T acting on the cable unit is calculated and obtained respectively. g Damping force D g Buoyancy B g and gravity G g The load is calculated using the following formula (2):

[0056]

[0057] In the above formula, g is the acceleration due to gravity, ε is the axial strain of element i, and d c Let E be the cable diameter, and c be Young's modulus and damping coefficient, respectively. l and l0 represent the current length and the initial length of the cable element before deformation, respectively.

[0058] In step S4, based on the calculated loads of all cable units and the mass of all cable units, the load and mass of each cable unit are evenly distributed to the nodes at both ends of the cable unit. The loads and masses acting on the nodes are substituted into the motion equations of the cable to complete the modeling and obtain the underwater cable dynamic model. The final dynamic equations of the cable unit nodes are shown in equation (3) below:

[0059]

[0060] In the above formula, Φ is the constraint vector. qLet Φ be the Jacobian matrix of Φ in terms of q, λ be the Lagrange multipliers, and q be the coordinate vector of all nodes. Represents the velocity vector of all nodes. This represents the acceleration vector of all nodes.

[0061] Existing methods such as the Euler method, Frenet method, relative velocity element method, and the global lumped mass method proposed in this embodiment differ in their representation of element mass and force. The Euler method uses Euler angles to define the local coordinate system; the Frenet method defines the local coordinate system based on the shape of the cable; the relative velocity element method determines the local coordinate system by the relative velocity between the cable element and the seawater; while the global lumped mass method has no local coordinate system, and its mass matrix and forces are all represented using the global coordinate system. Mathematical derivation confirms that the relative velocity element method and the global lumped mass method are actually equivalent; therefore, both methods have the same accuracy. However, the relative velocity element method consumes more computational resources than the global lumped mass method. The Euler method, Frenet method, relative velocity element method, and global lumped mass method are described in detail below.

[0062] (1) Euler method

[0063] like Figure 4 As shown, the rotation angles of the ZYX coordinate system are 0, θ, and θ respectively. The transformation matrix from the local coordinate system to the global coordinate system is shown in equation (4):

[0064]

[0065] In the above formula, A is the transformation matrix of element i.

[0066] θ and y are represented by the position vectors of node i and node i+1 in the global coordinate system. As shown in equation (5):

[0067]

[0068] In the above formula, r i x r i y r i z These are the x, y, and z components of the node position in the global coordinate system, respectively.

[0069] The forces acting on element i are divided into external forces and internal forces. The internal forces consist of tension and damping forces. Tension is related to axial strain, and damping force is related to the rate of change of axial strain, as shown in equation (6):

[0070]

[0071] Among them, Eg Let z be the element direction vector from node i to node i+1; l and l0 represent the current length and initial length of the cable element before deformation, respectively; g r is the unit z-vector representing element i in the global coordinate system. i and These are the position and velocity vectors of node i relative to the global coordinate system, respectively; ε is the axial strain of element i, and d... c denoted as cable diameter; E and c are Young's modulus and damping coefficient, respectively.

[0072] The external forces acting on the unit consist of gravity, resistance, and additional mass inertial forces, as shown in equation (7):

[0073]

[0074] Among them, v e It is the relative velocity of seawater (i.e., the fluid medium) with respect to the current cable element in the element coordinate system. v e Components in the x, y, and z directions; v w The velocity of the seawater in the global coordinate system; F D F ag G g and B g These are, respectively, drag, added mass inertial force, gravity, and buoyancy relative to the cable element in global coordinates; C f and C n These are the tangential drag coefficient and the normal drag coefficient, respectively; C m It is the additional mass coefficient; ρ c and ρ f The density of the cable and seawater; a w is the acceleration of seawater; g is the acceleration due to gravity.

[0075] The generalized mass matrix of cable element i relative to the global coordinate system is shown in equation (8).

[0076]

[0077] (2) Frenet Method

[0078] The difference between Euler and Frenet coordinate systems lies in how the three orthogonal bases are defined. Euler coordinates are defined using Euler angles, while Frenet coordinates define the orthogonal bases using the shape of the entire cable, such as... Figure 5 As shown.

[0079] In the global coordinate system, all node positions form three spline functions from the starting point to the node points. These three functions are named fi, fj, ... X fY and f Z Then the global coordinates of point P can be calculated using equation (9):

[0080]

[0081] In the above formula, X p Y p and Z p These represent the global coordinates of point P; These are the arc coordinates of point P.

[0082] A virtual node with coordinates r is set in the middle of each unit. v It can be obtained from equation (9).

[0083] Use node position r i r i+1 and virtual node position r v Define vectors α and β as shown in equation (10):

[0084]

[0085] The unit vectors t, n, and b in the tangential, normal, and binormal directions are shown in equation (11):

[0086]

[0087] Among them, I 3×3 It is a third-order identity matrix.

[0088] Therefore, the transformation matrix A can be written as:

[0089] A = [tnb] (12)

[0090] (3) Relative velocity element method

[0091] The element coordinate system is defined using the tangential vector and relative velocity in the relative velocity element coordinate system, and its three orthogonal bases are as shown in equation (13) and Figure 6 As shown:

[0092]

[0093] Therefore, the transformation matrix A can be written as:

[0094] A = [x g y g z g (14)

[0095] According to the relative velocity element coordinate system, the resistance on element i is:

[0096]

[0097] Furthermore, to verify the accuracy of the global lumped mass method, this embodiment compares the simulation results of the relative velocity element method, the global lumped mass method, and the Euler method (the Frenet method is singular here). The initial position of the cable is as follows: Figure 7 As shown in (a) above, the trajectory of the top node is as follows: Figure 7 As shown in (b) above, the positions of X, Y, and Z relative to time are as follows: Figure 7 As shown in (c) to (e), the detailed parameters of the relative velocity element method, the global lumped mass method, and the Euler method are shown in Table 1 below.

[0098] Table 1. Parameters of the relative velocity element method, the global lumped mass method, and the Euler method.

[0099]

[0100] The comparison results show that, at the same level of accuracy, the global lumped mass method is more efficient than the relative velocity element method and the Euler method.

[0101] Furthermore, the singularities of different coordinate system methods are analyzed, specifically the singularities of the Euler method, Frenet method, relative velocity element method, and global lumped mass method, further confirming that the global lumped mass method proposed in this embodiment does not exhibit singularities. Due to the properties of inverse trigonometric functions, when θ and For certain special angles, the coordinate system of the Euler method is singular, and the range of singularities is as follows: Figure 8 As shown in Table 2, the transformation matrix A is invalid within these singular ranges.

[0102] Table 2. Singularity range of the Euler coordinate system

[0103]

[0104] When all nodes are on a straight line, the coordinate system of the Frenet method is singular. In this case, the two nodes of the element and the virtual node are collinear. As can be seen from equation (11), the normal vector n' is zero at this time, so the normal unit vector n is meaningless, and thus the transformation matrix A is also invalid.

[0105] relative velocity v g and the tangential unit vector z g When parallel, the coordinate system of the relative velocity element method becomes singular, at which point x g and y g Meaningless.

[0106] When the length of element i is 0, the global lumped mass method may be singular, but in cable dynamics, the element length is not 0, so the global lumped mass method is not singular. Therefore, the singularities of the above four methods are summarized in Table 3 below.

[0107] Table 3. Singularity range of the four methods

[0108]

[0109] As another implementation, this embodiment proposes an arbitrary Lagrange-Eulerian global method model combining the arbitrary Lagrange-Eulerian method and the global lumped mass method to simulate the modeling of a variable-length flexible cable, which is a flexible cable whose length varies with time. The modeling of the variable-length flexible cable includes the following steps:

[0110] Using the steps of the cable dynamics modeling method based on the global lumped mass method described above, the internal and external forces acting on the nodes at both ends of the cable element are obtained;

[0111] Obtain the position coordinates of any node in the cable unit in the global coordinate system, and thus obtain the force related to the velocity of the material acting on the nodes at both ends of the cable unit;

[0112] The final calculation result is obtained by superimposing all the forces acting on the nodes at both ends of the cable unit;

[0113] Based on the final calculation results of all cable elements, the results are substituted into the motion equation of the cable. At the same time, based on the position coordinates of any node, additional constraint equations are derived using the arbitrary Lagrange-Euler method. The motion equations are updated in this way, the modeling is completed, and the dynamic model of the variable length cable in the fluid medium is obtained.

[0114] Specifically, such as Figure 9 As shown, to represent the change in cable length, a material coordinate p is added to each cable unit. Then, the coordinate q of unit i is:

[0115]

[0116] Let a variable s represent the node position in cell i, as follows:

[0117]

[0118] The position vector of any node in unit i is:

[0119]

[0120] In the above parameters, the subscripts i and i+1 represent the i-th node and the (i+1)-th node, respectively, i.e., p i p i+1 Let r be the material coordinates of the i-th node and the (i+1)-th node relative to the global coordinate system. i r i+1 Let be the position vectors of the i-th node and the (i+1)-th node relative to the global coordinate system.

[0121] Based on the position vector of any node in the cable element in the global coordinate system, the velocity vector, acceleration vector and axial strain of any node in the cable element are calculated respectively, which lays the foundation for subsequent calculation of the forces related to the material velocity acting on the nodes at both ends of the cable element.

[0122] Specifically, the velocity vector of any node in unit i is:

[0123]

[0124] in, It is the velocity vector of the node relative to the global coordinate system.

[0125] The acceleration vector of any node in element i is:

[0126]

[0127] in, This represents the acceleration of the node relative to the global coordinate system.

[0128] The axial strain at any node in element i can be written as equation (21):

[0129]

[0130] Where ε0 is the axial strain.

[0131] Based on the force acting on element i, the forces acting on node i and node i+1 are divided into three parts, as shown in equation (22):

[0132]

[0133] In the above formula, and Let i be the internal force and the external force at node i, respectively. The forces at nodes i and i+1 are related to the velocity of the material. When the cable length is constant... If the mass is zero, then any Lagrange-Euler global method is a global lumped mass method. That is, there exists... Only then can variable-length cables be represented using arbitrary Lagrange-Euler global methods.

[0134] Additional inertial generalized force and Represented as:

[0135]

[0136] Internal forces acting on element i and It can be obtained by calculation from equation (24):

[0137]

[0138] external force and It can be derived from equation (25) as follows:

[0139]

[0140] In the above formula, f is the external force acting on the infinitesimal element, and its calculation formula is shown in formula (26):

[0141]

[0142] The mass matrices of node i and node i+1 can be represented as:

[0143]

[0144] The additional mass matrix of element i is written as:

[0145] M add =C m ρ f AP (28)

[0146] The arbitrary Lagrange-Euler global method contains some additional constraint equations derived from the Lagrange-Euler method. These equations are related to the material coordinates of all nodes and are as follows:

[0147]

[0148] In the above formula, f is a function that calculates these constraints.

[0149] By superimposing all the forces acting on the nodes at both ends of the cable unit obtained from the above calculations, the final calculation result is obtained. Based on the final calculation results of all cable units, the results are substituted into the motion equation of the cable. At the same time, based on the position vector of any node, additional constraint equations are derived using arbitrary Lagrange-Euler methods. The motion equations are updated in this way, the modeling is completed, and the dynamic model of a variable-length cable in a fluid medium is obtained.

[0150] Furthermore, the feasibility of the above-described scheme in this embodiment is further confirmed through the following simulation. The arbitrary Lagrange-Eulerian description-absolute nodal coordinate method is a finite element method that can establish dynamic models of beams with varying lengths. It uses the position coordinates, gradient coordinates, and material coordinates of all nodes to represent the dynamic equations. These coordinates can be expressed as:

[0151]

[0152] In the above formula, and These are the gradient coordinates of node i and node i+1, respectively.

[0153] Furthermore, the arbitrary Lagrange-Euler global method and the arbitrary Lagrange-Euler description-absolute nodal coordinate method differ in their expressions for internal forces. The latter, due to the presence of gradient coordinates, more accurately represents the dynamic model of a bending beam than the former, and its internal force equations, including bending forces, are shown in equation (31):

[0154]

[0155] In the above formula, κ is the curvature of the node corresponding to the material coordinate p, and J is the moment of inertia of the cross section.

[0156] On the other hand, because the arbitrary Lagrange-Euler description-absolute nodal coordinate method contains additional coordinates (gradient coordinates), its dynamic equations are more numerous than those of the arbitrary Lagrange-Euler global method, thus resulting in a longer simulation time. Considering the relatively small bending force, the arbitrary Lagrange-Euler global method was chosen for cable modeling. The simulation results of the two methods are as follows.

[0157] Furthermore, this simulation employs an adaptive method for adjusting the number of nodes, with the initial positions of all nodes as follows: Figure 10 As shown in (a); the trajectory of the top cable node is circular, as... Figure 10 As shown in (b); the cable length is as shown in Figure 1. Figure 10 As shown in (c); the dynamic response of the cable, i.e., the X, Y, Z coordinates of the tail node, is as follows: Figure 10 As shown in (d), (e), and (f); the parameters of ALE-GLMF and ALE-ANCF are shown in Table 4 below.

[0158] Table 4. Parameters of the Arbitrary Lagrange-Eulerian Global Method and the Arbitrary Lagrange-Eulerian Description-Absolute Nodal Coordinate Method

[0159]

[0160] To address the singularity problem inherent in traditional lumped mass methods for modeling slender cables, this embodiment proposes a global lumped mass method that utilizes only the global motion of cable nodes to establish slender cable models. Furthermore, for variable-length cable models widely used in marine towed systems, an arbitrary Lagrange-Eulerian global method model combining the arbitrary Lagrange-Eulerian method and the global lumped mass method is proposed. The methods described in this embodiment can be applied to marine towed systems, fishing vessels, and other similar applications, providing valuable reference for the further application and development of marine towed systems. Similarly, the methods described in this embodiment can also be applied to the deployment of tower cranes, aerial refueling hoses, cables, and pipelines, offering valuable reference for these applications as well.

[0161] Example 2

[0162] This embodiment provides a dynamic modeling system for slender cables based on the global lumped mass method, including:

[0163] The global position coordinate acquisition module is used to acquire the shape of the flexible slender cable in the fluid medium in the global coordinate system. The cable is divided into multiple cable units by using the lumped mass method. The position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium are acquired.

[0164] The calculation module is used to obtain the tangential vector and normal plane of the cable unit based on the position coordinates of the nodes at both ends of the cable unit. It then projects the relative velocity vector into the tangential and normal components of the relative velocity using the tangential vector and normal plane, thereby determining the unit tangential vector and unit normal vector of the relative velocity. Finally, it obtains the unit binormal vector of the relative velocity according to the right-hand rule, and uses this to calculate the fluid resistance, lift, and additional mass load acting on the cable unit. Based on the position coordinates of the nodes at both ends of the cable unit, it obtains the tension, damping force, buoyancy, and gravity load acting on the cable unit.

[0165] The modeling module is used to substitute the calculation results of all cable elements into the motion equation of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0166] Furthermore, based on the calculated loads of all cable units and their masses, the loads and masses of each cable unit are evenly distributed to the nodes at both ends of the cable unit. The loads and masses acting on the nodes are then substituted into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

[0167] Example 3

[0168] This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the processor executes the computer instructions, it completes the steps in the slender cable dynamics modeling method based on the global lumped mass method as described above.

[0169] Example 4

[0170] This embodiment also provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps in the slender cable dynamics modeling method based on the global lumped mass method as described above.

[0171] The steps and methods involved in Embodiments 2 to 4 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0172] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0173] The above description is only a preferred embodiment of the present invention. Although the specific embodiments of the present invention have been described in conjunction with the accompanying drawings, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that, based on the technical solutions of the present invention, various modifications or variations that can be made by those skilled in the art without creative effort are still within the scope of protection of the present invention.

Claims

1. A method for modeling the dynamics of slender cables based on the global lumped mass method, characterized in that, include: In the global coordinate system, the shape of the flexible slender cable in the fluid medium is obtained. The cable is divided into multiple cable units by the lumped mass method. The position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium are obtained. Based on the position coordinates of the nodes at both ends of the cable unit, the tangential vector and normal plane of the cable unit are obtained. The relative velocity vector is projected into the tangential component and normal component of the relative velocity using the tangential vector and normal plane. Then, the unit tangential vector and unit normal vector of the relative velocity are determined. The unit binormal vector of the relative velocity is obtained according to the right-hand rule. In this way, the fluid resistance, lift and additional mass load acting on the cable unit are calculated. Based on the position coordinates of the nodes at both ends of the cable unit, the tension, damping force, buoyancy and gravity loads acting on the cable unit are obtained; Based on the calculation results of all cable elements, the results are substituted into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

2. The global lumped-mass based modeling method for the dynamics of an elongated cable of claim 1, wherein, The calculation results of all cable units are substituted into the motion equations of the cable to complete the modeling, resulting in a cable dynamics model in the fluid medium, including: Based on the calculated loads of all cable units and their masses, the loads and masses of each cable unit are evenly distributed to the nodes at both ends of the cable unit. The loads and masses acting on the nodes are then substituted into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

3. The global lumped-mass based modeling method for the dynamics of an elongated cable of claim 1, wherein, The forces distributed on each cable unit and its two end nodes include tension, damping force, gravity, buoyancy and hydrodynamic force. The forces acting on the cable unit are divided into internal forces and external forces. Internal forces include tension and damping force, while external forces include gravity, buoyancy and hydrodynamic force.

4. The method for modeling the dynamics of slender cables based on the global lumped mass method as described in claim 1, characterized in that, Based on the determined unit tangent vector and unit normal vector of the relative velocity, the unit binormal vector of the relative velocity is obtained using the right-hand rule. The normal fluid resistance, tangential fluid resistance, vertical lift, and normal additional mass load acting on the cable element are then calculated.

5. A global lumped-mass based modeling method of the dynamics of an elongated cable as claimed in any one of claims 1 to 4, characterized in that, A global method combining arbitrary Lagrange-Eulerian method and global lumped mass method is used to model a variable-length flexible cable in a fluid medium. The variable-length flexible cable is a flexible cable whose length changes over time.

6. The global lumped-mass based modeling method of the dynamics of an elongate cable of claim 5, wherein, Modeling of variable-length flexible cables in fluid media includes: A cable dynamics modeling method based on the global lumped mass method is adopted to obtain the internal and external forces acting on the nodes at both ends of the cable element; Obtain the position coordinates of any node in the cable unit in the global coordinate system, and thus obtain the force related to the velocity of the material acting on the nodes at both ends of the cable unit; The final calculation result is obtained by superimposing all the forces acting on the nodes at both ends of the cable unit; Based on the final calculation results of all cable elements, the results are substituted into the motion equation of the cable. At the same time, based on the position coordinates of any node, additional constraint equations are derived using the arbitrary Lagrange-Euler method. The motion equations are updated in this way, and the modeling is completed, resulting in a dynamic model of a variable-length cable in a fluid medium.

7. An elongated cable dynamics modeling system based on a global lumped mass method, characterized by, include: The global position coordinate acquisition module is used to acquire the shape of the flexible slender cable in the fluid medium in the global coordinate system. The cable is divided into multiple cable units by using the lumped mass method. The position coordinates of the two ends of each cable unit in the global coordinate system and the relative velocity vector of the cable unit in the fluid medium are acquired. The calculation module is used to obtain the tangential vector and normal plane of the cable unit based on the position coordinates of the nodes at both ends of the cable unit. It then uses the tangential vector and normal plane to project the relative velocity vector into the tangential component and normal component of the relative velocity, thereby determining the unit tangential vector and unit normal vector of the relative velocity. Finally, it obtains the unit binormal vector of the relative velocity according to the right-hand rule, and uses this to calculate the fluid resistance, lift and additional mass load acting on the cable unit. Based on the position coordinates of the nodes at both ends of the cable unit, the tension, damping force, buoyancy and gravity loads acting on the cable unit are obtained; The modeling module is used to substitute the calculation results of all cable elements into the motion equation of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

8. The global lumped-mass based, slender-cable dynamics modeling system of claim 7, wherein, The calculation results of all cable units are substituted into the motion equations of the cable to complete the modeling, resulting in a cable dynamics model in the fluid medium, including: Based on the calculated loads of all cable units and their masses, the loads and masses of each cable unit are evenly distributed to the nodes at both ends of the cable unit. The loads and masses acting on the nodes are then substituted into the motion equations of the cable to complete the modeling and obtain the cable dynamics model in the fluid medium.

9. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, complete the steps of a slender cable dynamics modeling method based on the global lumped mass method as described in any one of claims 1-6.

10. A computer readable storage medium characterized by, Used to store computer instructions, which, when executed by a processor, complete the steps of a slender cable dynamics modeling method based on the global lumped mass method as described in any one of claims 1-6.

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