A rigid-flexible coupling bidirectional closed-loop time-domain dynamics calculation method for underwater towed system

Abstract

The application discloses a kind of rigid-flexible coupling two-way closed loop time domain dynamics calculation methods of underwater towing system, belong to the field of ocean engineering and underwater equipment technology;Method includes: establishing multi-coordinate system framework, constructs the six-degree-of-freedom rigid body motion equation of underwater vehicle, buoy and the partial differential dynamics equation of continuous elastic body of tow cable, is integrated into whole coupling model by displacement, velocity and force boundary condition.Set up two-way closed loop calculation framework: in each time step, first solve the boundary force of two rigid bodies based on the state of tow cable, and parallelly use four-order Runge-Kutta method to advance rigid body state simultaneously;Again, the updated rigid body position and velocity are used as forced boundary condition, and the tow cable is solved again using mixed difference method, to obtain the coordinated cable shape and tension distribution at the next time.This application realizes the dynamic synchronous update of rigid body motion and flexible cable deformation, and overcomes the phase lag and amplitude error of traditional weak coupling method.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering and underwater equipment technology, specifically relating to a rigid-flexible coupled two-way closed-loop time-domain dynamics calculation method for underwater towing systems. Background Technology

[0002] Underwater towed systems are indispensable key equipment in fields such as marine resource exploration and hydrological surveys. Their performance and safety are highly dependent on the complex dynamic behavior of the system under the combined effects of wind, waves, and currents. Therefore, developing dynamic calculation methods that can accurately predict the system's time-domain response has significant engineering and scientific value.

[0003] Currently, the dynamic analysis methods for underwater towing systems can be mainly classified into the following categories: 1. Lumped Mass Method: This method discretizes the towed cable into a series of mass points connected by springs and dampers, neglecting its bending stiffness. Its advantages include a relatively simple model and high computational efficiency. However, this method struggles to accurately describe the continuous medium characteristics of the towed cable, the continuous distribution of internal tension, and wave propagation effects. It has inherent limitations in handling large deformations, dynamic impacts, and high-precision tension prediction, and is particularly difficult to achieve high-fidelity coupling with the six-degree-of-freedom rigid body motion of underwater vehicles and buoys.

[0004] 2. Finite Element Method: This method models the towed cable as beam or cable elements, accurately considering its geometric nonlinearity, material nonlinearity, and bending stiffness, theoretically offering the highest accuracy. However, its drawback is the extremely large computational load, which is often prohibitive for underwater towed systems requiring long simulations and containing multiple rigid body components. Furthermore, the highly nonlinear dynamic equations and rigidity characteristics pose a significant challenge to the stability of numerical integration algorithms, easily leading to computational divergence or requiring extremely small time steps, thus limiting its practicality.

[0005] 3. Quasi-static methods: Some studies simplify the tow cable into a steady-state catenary model to reduce complexity, or ignore certain degrees of freedom and only perform dynamic analysis on the vehicle. These methods greatly sacrifice dynamic accuracy and cannot simulate the transient impact, resonance, and strongly nonlinear dynamic processes caused by emergency maneuvers under wave excitation.

[0006] 4. Weak Coupling or Sequential Solution Strategies: While some existing technologies have proposed coupled modeling approaches for surface vessels (rigid bodies), towlines (continuum), and towed bodies (rigid bodies), and employ fourth-order Runge-Kutta methods and finite difference methods respectively, their coupling methods are often loose sequential iterations or unidirectional transfer of boundary conditions. That is, within each time step, the vessel motion may be solved first, then the result used as fixed boundary conditions to solve the towline, and finally the towed body. This method severs the real-time, bidirectional dynamic interaction between the system components, failing to capture the essential feedback between force and motion. Especially under high-frequency wave excitation or when the system undergoes severe maneuvers, significant phase lag and amplitude errors occur, leading to distorted predictions of key phenomena such as the peak dynamic tension of the towline and the instability of the buoy / vehicle.

[0007] In summary, existing technologies for underwater towed system dynamics calculations generally face the dilemma of "difficulty in simultaneously achieving accuracy, efficiency, and coupling integrity." In particular, there is a lack of a high-fidelity time-domain dynamics calculation method that can realize bidirectional closed-loop data exchange and ensure dynamic coupling and synchronous updates of each component in every time step. Summary of the Invention

[0008] The technical problem to be solved: To avoid the shortcomings of existing technologies, this invention provides a rigid-flexible coupled bidirectional closed-loop time-domain dynamics calculation method for underwater towed systems. The core of this method is to construct a closed-loop calculation framework with strict temporal logic and bidirectional data exchange to ensure that the motion states of the three core components—the underwater vehicle, the tow cable, and the buoy—are dynamically coupled and synchronously updated in each time step, thereby accurately simulating the nonlinear transient response of the system under complex sea conditions.

[0009] The technical solution of this invention is: a rigid-flexible coupled two-way closed-loop time-domain dynamics calculation method for underwater towing systems, the specific steps of which are as follows: Step 1: Establish a multi-coordinate system framework: Construct multiple coordinate systems to describe the spatial motion of the underwater vehicle, flexible tow cable, and surface buoy, including a global fixed coordinate system, a follower coordinate system fixed to the underwater vehicle, a follower coordinate system fixed to the buoy, and a local coordinate system to describe the local shape and force of the tow cable. Step 2: Establish independent motion control equations for components: Treat the underwater vehicle and buoy as rigid bodies and establish their six-degree-of-freedom motion equations respectively; treat the tow cable as a continuous elastic body and establish its dynamic partial differential equations, which include fluid resistance, inertial force, gravity, buoyancy and time-varying load terms of internal tension. Step 3: Construct an overall coupled dynamic model: By using displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions, the motion control equations of the underwater vehicle, tow cable, and buoy are coupled at the connection boundary to form an overall dynamic model that reflects the dynamic interaction between system components. Step 4: Construct a closed-loop computing framework for bidirectional data exchange for collaborative solving: Within each time step, execute the following sub-steps in strict timing order: S1. Coupled boundary force transmission: Based on the shape and velocity field of the tow cable at the current moment, solve the motion control equation of the tow cable to obtain the time-varying tension and torque acting on the underwater vehicle at the tow cable head end, and the time-varying tension and torque acting on the buoy at the tow cable tail end. S2. Rigid body state cooperative propulsion: The boundary forces and moments obtained in S1 are used as external loads and substituted into the six-degree-of-freedom motion equations of the underwater vehicle and the buoy respectively. The ordinary differential equations of each are integrated in parallel over time to predict the position, attitude and velocity of the underwater vehicle and the buoy at the next moment. S3. Synchronous Update of Tow Cable Status: The position and velocity of the underwater vehicle and buoy at the next moment predicted by S2 are used as the updated imposed boundary conditions. The motion control equation of the tow cable is solved again to obtain the overall shape, node velocity and tension distribution of the tow cable at the next moment that are consistent with the updated boundary. S4. System State Synchronization and Looping: Summarize the next-time states of all components obtained in S2 and S3, update the complete state vector of the entire coupled system, and proceed to the iterative calculation of the next time step; Step 5: Obtain the system dynamic response: Through the above closed-loop collaborative solution, obtain the complete dynamic response of the underwater towed system in the time domain, including the position, attitude, and velocity of the underwater vehicle and buoy, as well as the spatial configuration, tension distribution, and dynamic load history of the tow cable.

[0010] A further technical solution of the present invention is: the independent motion control equations of each component in step two are specifically as follows: The motion control equations for underwater vehicles are established based on the theorems of momentum and angular momentum, and their form is as follows:

[0011] in, Let be the inertial matrix of the underwater vehicle in volume coordinates. The Coriolis force-centrifugal force matrix represents the force of an underwater vehicle moving in volume coordinates. and Let these be the generalized velocity vector and generalized acceleration vector of the underwater vehicle in the body coordinate system. It is the resultant force vector of all external forces and external moments acting on the underwater vehicle.

[0012] The motion control equation for the towing cable is based on d'Alembert's principle, using the tension on the towing cable... Local velocity components , , and attitude angle , A system of partial differential equations established for the variables; Buoy motion control equations: adopting a six-degree-of-freedom rigid body motion equation similar to that of a ship, with external force terms including gravity, buoyancy, wave force, and tension and torque transmitted from the tail end of the tow cable. A further technical solution of the present invention is: the coupling boundary condition in step three is specifically as follows: Bow boundary condition: The velocity at the bow of the tow cable is the same as the velocity at the tow point on the underwater vehicle, satisfying the condition.

[0013] In the formula, The translational speed of the underwater vehicle. The rotational angular velocity of the underwater vehicle. This is the position vector of the tow cable tip in the underwater vehicle's follower coordinate system; This is the transformation matrix between the fixed spatial coordinate system and the underwater vehicle's moving coordinate system. The transformation matrix between the fixed spatial coordinate system and the local coordinate system of the towing cable. The velocity of the tow cable head in the tow cable local coordinate system.

[0014] Tail-end boundary condition: The velocity at the tail end of the tow cable is the same as the velocity at the tow point on the buoy, satisfying the condition.

[0015] In the formula, Let be the buoy's translational speed. Let be the angular velocity of the buoy. is the position vector of the tow cable tail end in the buoy's following coordinate system; This is the transformation matrix between the buoy's moving coordinate system and the fixed spatial coordinate system. The transformation matrix between the fixed spatial coordinate system and the local coordinate system of the towing cable. The velocity of the trailing cable end in the local coordinate system of the trailing cable.

[0016] A further technical solution of the present invention is: in S2, the six-degree-of-freedom motion equations of the underwater vehicle and the buoy are time-integrated using the fourth-order Runge-Kutta method, and the two are calculated in parallel.

[0017] A further technical solution of the present invention is as follows: In S1 and S3, the motion control equations of the tow cable are solved using the hybrid difference method. Specifically, the spatial derivative terms of the tow cable are discretized using the central difference scheme, and the partial differential equations are transformed into ordinary differential equations about the spatial discrete node variables. Then, the implicit time integration scheme is used to iteratively solve the ordinary differential equations.

[0018] A further technical solution of the present invention is that the implicit time integral in the hybrid difference method is solved by implicit Euler method or Newton-Raphson iteration to solve the nonlinear algebraic equation system to ensure numerical stability.

[0019] A further technical solution of the present invention is: in the closed-loop calculation framework of step four, the sequential process from S1 to S4 is executed only once within each time step, without internal sub-iteration, and the consistency of the coupling state is achieved through the implicit time integration of the cable solver in S3.

[0020] A further technical solution of the present invention is that the dynamic response obtained in step five includes: the six-degree-of-freedom motion history of the underwater vehicle and the buoy, the spatial position and velocity history of each node of the tow cable, the tension distribution history of the tow cable along the arc length, and the dynamic tension and torque history of the tow cable head and tail ends. A further technical solution of the present invention is: in the multi-coordinate system framework, the transformation between each coordinate system is achieved by the Euler angle direction cosine matrix, wherein the transformation matrix between the following coordinate system of the underwater vehicle and the buoy and the fixed spatial coordinate system is determined by the roll angle, pitch angle and yaw angle.

[0021] A rigid-flexible coupled two-way closed-loop time-domain dynamics simulation system for an underwater towing system includes: The model building unit constructs a rigid body six-degree-of-freedom motion model for an underwater vehicle, a rigid body six-degree-of-freedom motion model for a buoy, and a partial differential dynamics model for a towed cable continuous elastic body, and establishes coordinate system transformation relationships and physical parameter mappings between the models. The coupled boundary condition management unit defines the displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions between the underwater vehicle and the tow cable head end, and between the tow cable tail end and the buoy, as constraints for data exchange between the various models. The rigid body motion solver uses the fourth-order Runge-Kutta method to perform time integration on the six-degree-of-freedom motion ordinary differential equations of underwater vehicles and buoys, and outputs the position, attitude and velocity of the rigid body at the next moment. The flexible cable solver uses the hybrid finite difference method to spatially discretize and implicitly integrate the partial differential dynamic equations of the tow cable, and outputs the spatial position, velocity and tension distribution along the cable at each node. A bidirectional closed-loop coupled controller is connected to the rigid body motion solver, the flexible cable solver, and the coupled boundary condition management unit, and performs the following operations sequentially within each time step: The flexible cable solver is triggered to calculate the boundary tension and moment at the head and tail ends based on the current tow cable state; The boundary tension and torque are input as external loads to the rigid body motion solver, driving it to advance the rigid body state of the underwater vehicle and the buoy in parallel. The new position and velocity output by the rigid body solver are used as updated forced boundary conditions to trigger the flexible cable solver to synchronously correct the tow cable state. Collect the output results of the rigid body solver and the flexible cable solver to form the complete system state vector at the next moment; The system state memory stores the complete state sequence of the underwater vehicle, buoy, and tow cable generated by the bidirectional closed-loop coupled controller in time steps. The simulation output interface reads and outputs time-domain dynamic response data from the system state memory, including the motion trajectory and attitude history of the rigid body, the spatial configuration evolution of the tow cable, and the tension distribution history.

[0022] Beneficial effects The beneficial effects of this invention are as follows: (1) Achieved bidirectional dynamic coupling: By constructing a closed-loop computing framework with strict temporal logic and bidirectional data exchange, this invention overcomes the information lag problem of sequential solution or weak coupling methods in the prior art, and ensures the dynamic coupling and synchronous update of rigid body motion and flexible cable deformation in each time micro-element, laying the methodological foundation for high-precision time domain analysis.

[0023] (2) Significantly improves the prediction accuracy of system dynamic response under complex sea conditions: The multi-coordinate system coupled dynamic model established in this invention fully incorporates the six degrees of freedom motion of the underwater vehicle, tow cable, and buoy. In particular, by accurately solving the partial differential equations of the tow cable through a closed-loop computational framework, the dynamic characteristics of the cable can be simulated more realistically. Comparative analysis shows that compared with the sequential solution method, this invention improves the prediction accuracy of the dynamic tension peak of the tow cable by about 15% and effectively eliminates phase error.

[0024] (3) Enhanced numerical computation stability: The present invention achieves bidirectional coupling through implicit iteration in each time step, which significantly enhances the numerical stability of the algorithm. Under complex sea conditions, when the traditional sequential method terminates due to computational divergence, the method of the present invention can still stably complete the full time-domain simulation.

[0025] (4) Provides a more practical and complete system simulation solution: This invention couples the six-degree-of-freedom dynamic response of the buoy, the continuous medium model of the tow cable and the motion model of the underwater vehicle through strict physical boundary conditions and closed-loop calculation framework. Within the same framework, it realizes high-fidelity time-domain simulation of the entire system from the surface to underwater and from rigid body to flexible body, providing a high-confidence analysis platform for system-level design parameter optimization, safe operation window assessment and risk prediction under extreme conditions. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the basic process of a rigid-flexible coupled bidirectional closed-loop time-domain dynamics calculation method for an underwater towing system according to an embodiment of the present invention.

[0027] Figure 2 This is a schematic diagram of an underwater towing system model, illustrating a rigid-flexible coupling bidirectional closed-loop time-domain dynamics calculation method for an underwater towing system according to an embodiment of the present invention. The diagram shows the coordinate systems and connection points.

[0028] Figure 3 This is a time curve showing the change in tension at the tow cable head end in an embodiment of the present invention, which compares and shows the difference between the method of the present invention and the sequential solution method.

[0029] Figure 4 This describes the evolution of the tow cable spatial configuration over time in the embodiments of the present invention. Detailed Implementation

[0030] The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0031] To address the problems existing in current technologies, this invention proposes a rigid-flexible coupled two-way closed-loop time-domain dynamics calculation method for underwater towing systems. The specific steps are as follows: Step 1: Establish a multi-coordinate system framework: Construct multiple coordinate systems to describe the spatial motion of the underwater vehicle, flexible tow cable, and surface buoy, including a global fixed coordinate system, a follower coordinate system fixed to the underwater vehicle, a follower coordinate system fixed to the buoy, and a local coordinate system to describe the local shape and force of the tow cable. Step 2: Establish independent motion control equations for components: Treat the underwater vehicle and buoy as rigid bodies and establish their six-degree-of-freedom motion equations respectively; treat the tow cable as a continuous elastic body and establish its dynamic partial differential equations, which include fluid resistance, inertial force, gravity, buoyancy and time-varying load terms of internal tension. Step 3: Construct an overall coupled dynamic model: By using displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions, the motion control equations of the underwater vehicle, tow cable, and buoy are coupled at the connection boundary to form an overall dynamic model that reflects the dynamic interaction between system components. Step 4: Construct a closed-loop computing framework for bidirectional data exchange for collaborative solving: Within each time step, execute the following sub-steps in strict timing order: S1. Coupled boundary force transmission: Based on the shape and velocity field of the tow cable at the current moment, solve the motion control equation of the tow cable to obtain the time-varying tension and torque acting on the underwater vehicle at the tow cable head end, and the time-varying tension and torque acting on the buoy at the tow cable tail end. S2. Rigid body state cooperative propulsion: The boundary forces and moments obtained in S1 are used as external loads and substituted into the six-degree-of-freedom motion equations of the underwater vehicle and the buoy respectively. The ordinary differential equations of each are integrated in parallel over time to predict the position, attitude and velocity of the underwater vehicle and the buoy at the next moment. S3. Synchronous Update of Tow Cable Status: The position and velocity of the underwater vehicle and buoy at the next moment predicted by S2 are used as the updated imposed boundary conditions. The motion control equation of the tow cable is solved again to obtain the overall shape, node velocity and tension distribution of the tow cable at the next moment that are consistent with the updated boundary. S4. System State Synchronization and Looping: Summarize the next-time states of all components obtained in S2 and S3, update the complete state vector of the entire coupled system, and proceed to the iterative calculation of the next time step; Step 5: Obtain the system dynamic response: Through the above closed-loop collaborative solution, obtain the complete dynamic response of the underwater towed system in the time domain, including the position, attitude, and velocity of the underwater vehicle and buoy, as well as the spatial configuration, tension distribution, and dynamic load history of the tow cable.

[0032] This invention proposes a two-way closed-loop time-domain dynamics calculation method for rigid-flexible coupling underwater towed systems, aiming to solve the problem of insufficient dynamic response prediction accuracy caused by the simplification of coupling relationships between components or sequential solution strategies in existing technologies. The method first establishes a multi-coordinate system framework and constructs independent motion control equations for the underwater vehicle, tow cable, and buoy. Through coupled boundary conditions of displacement coordination, velocity continuity, and force balance, the overall dynamic model of the underwater towed system is integrated. The core of this invention lies in constructing a closed-loop calculation framework with bidirectional data exchange: within each time step, the boundary forces of the tow cable are first solved based on its current state, and then input in parallel to the rigid body motion equations of the buoy and vehicle, using the fourth-order Runge-Kutta method to synchronously advance its state; subsequently, the updated rigid body state is used as the imposed boundary conditions, and the tow cable is solved again using the hybrid finite difference method to obtain the cable morphology and tension distribution at the next time step in coordination with it. Through this strictly time-sequential closed-loop iteration, dynamic coupling and synchronous updating of rigid body motion and flexible cable deformation are achieved. This invention effectively overcomes the phase lag and amplitude error of traditional weakly coupled methods, significantly improves the prediction accuracy and numerical stability of system dynamic response under complex sea conditions, and provides a high-fidelity theoretical tool for the design optimization and operational safety assessment of towed systems.

[0033] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0034] In one embodiment, refer to Figure 1 As shown, a rigid-flexible coupled two-way closed-loop time-domain dynamics calculation method for an underwater towed system includes the entire process from coordinate system establishment, component equation construction, overall coupled model formation, to finally obtaining the dynamic response through collaborative numerical solution. The specific steps are as follows: Step 1: Establish a multi-coordinate system framework: Ignore the influence of the tow cable and buoy system on the flow field around the underwater vehicle; the interaction between the underwater vehicle and the towed system is reflected through the forces transmitted by the tow cable. (Refer to...) Figure 2 As shown, the system consists of three parts: a surface buoy, a flexible tow cable, and an underwater vehicle. A multi-coordinate system framework is established to describe the spatial motion of each component of the system, including a globally fixed coordinate system. The servo coordinate system fixed to the underwater vehicle A local coordinate system used to describe the local shape and forces acting on the towing cable. and the follower coordinate system fixed to the buoy .like Figure 2 As shown, the tail end of the tow cable is connected to the towing point P on the buoy, and the bow end is connected to the towing point Q on the underwater vehicle. The coupling of force and motion is achieved through the connection points.

[0035] Step 2: Establish independent motion equations for each component: Independent motion control equations are constructed for the three core components: the underwater vehicle, the tow cable, and the buoy. The tow cable is treated as a continuous elastic body, and its dynamic equations fully consider various time-varying loads such as fluid resistance, inertial force, gravity, buoyancy, and internal tension. The underwater vehicle and the buoy are treated as rigid bodies, and six-degree-of-freedom motion equations are established for each.

[0036] Step 3: Construct an overall coupled dynamic model: By using displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions, the motion control equations of the above three components are coupled at the connection boundary to establish an overall coupled dynamic model of the underwater towing system that can accurately reflect the dynamic interaction between the system components.

[0037] Step 4: Construct a closed-loop computational framework with bidirectional data exchange to collaboratively solve the overall coupled dynamics model. This includes the following sub-steps: S1. Coupled Boundary Force Transmission Calculation: Based on the current shape and velocity field of the tow cable, solve the motion control equation of the tow cable, and calculate the time-varying tension and torque acting on the underwater vehicle at the tow cable head end, and the time-varying tension and torque acting on the buoy at the tow cable tail end. S2. Rigid body state cooperative propulsion: The boundary forces and moments calculated in S1 are used as external loads and substituted into the six-degree-of-freedom motion control equations of the underwater vehicle and the buoy respectively. The ordinary differential equations are integrated in parallel over time to predict the position and velocity of the underwater vehicle and the buoy at the next moment. S3. Synchronous Update of Tow Cable Status: The new position and velocity of the underwater vehicle and buoy obtained in S2 at the next moment are used as the updated imposed boundary conditions. Under the constraints of these new boundary conditions, the motion control equations of the tow cable are solved again to obtain the overall shape, node velocity and tension distribution of the tow cable at the next moment that are in harmony with the new boundary state. S4. System State Synchronization and Looping: Summarize the next-time states of all components calculated in S2 and S3, update the complete state vector of the entire coupled system, and then proceed to the iterative calculation of the next time step.

[0038] Step 5: Obtain the system dynamic response: Through the above collaborative solution process, obtain the complete dynamic response of the entire underwater towing system in the time domain, including the position and velocity of the underwater vehicle and buoy, as well as key physical quantities such as the spatial configuration and tension distribution of the towing cable.

[0039] In one embodiment, the specific method for establishing the independent motion control equations for each component in step two is as follows: Equations for motion control of underwater vehicles: Based on the momentum and angular momentum equations, the motion control equations of an underwater vehicle can be written as: (1) (2) (3) In the formula, Let be the inertial matrix of the underwater vehicle in volume coordinates. The Coriolis force-centrifugal force matrix represents the force of an underwater vehicle moving in volume coordinates. and Let these be the generalized velocity vector and generalized acceleration vector of the underwater vehicle in the body coordinate system. The mass of the underwater vehicle; This refers to the coordinates of the center of mass of an underwater vehicle in the volume coordinate system; , , It refers to the angular velocity component of the center of mass of an underwater vehicle in the body coordinate system; , , underwater vehicle body coordinate system , , Moment of inertia of the shaft , , , , , The inertial product of an underwater vehicle; This is the resultant force vector of all external forces and torques acting on the underwater vehicle. The force exerted by the tow cable on the underwater vehicle is calculated as a boundary condition coupled to the tow cable's motion equations.

[0040] Control equations for tow cable motion: Based on d'Alembert's principle, using six variables , , , , and To describe the motion of the tow cable, by simultaneously establishing and simplifying the dynamic balance equations and motion equations of the tow cable, we can obtain the motion control equations of the tow cable as follows: (4) In the formula, (5) (6) (7) (8) In the formula, The tension on the tow cable; Let the velocity of the tow cable be in the local coordinate system. This refers to the mass of the tow cable per unit length when it is not stretched. For fluid density, It is the acceleration due to gravity; ; and These are the tangential and normal drag coefficients, respectively. A This represents the cross-sectional area of ​​the tow cable when it is not stretched. In response to the situation; e=1 / EA , E The Young's modulus of the cable; , For the water flow and velocity in the direction; , For the water flow and Acceleration in the direction of; Let be the vertical velocity at a point on the tow cable; , . For a point on the tow cable relative to the water flow , and Velocity in a certain direction.

[0041] The position of the tow cable in a fixed spatial coordinate system is: (9) Buoy motion control equations: The buoy is a rigid body, and the mass of the buoy is... Buoy relative to moving coordinate system The moments of inertia in the three directions are respectively , and Buoy center of gravity The velocity in the moving coordinate system is (Longitudinal velocity, lateral velocity, and vertical velocity are respectively) The longitudinal acceleration, lateral acceleration, and vertical acceleration are respectively... ; angular velocity is (The roll rate, pitch rate, and yaw rate are respectively...) The roll acceleration, pitch acceleration, and yaw acceleration are respectively and The external force acting on the buoy is The torque of the external force about the center of gravity of the buoy is The equation of motion for the buoy in six-degree-of-freedom space can then be written as: (10) (11) (12) (13) (14) (15) The buoy is subjected to the tension of the tow cable transmitted from the tail end of the tow cable. and torque They are respectively (16) (17) In the formula, This is the expression for the tension at the tail end of the tow cable in the local coordinate system of the tow cable; is the position vector of the tow cable tail end in the buoy's following coordinate system; The transformation matrix between the drag body's moving coordinate system and the fixed spatial coordinate system is shown in equation (18).

[0042] (18) In the formula, , , These are the tow body's roll angle, pitch angle, and bow angle, respectively.

[0043] In one embodiment, the coupling boundary conditions in step three specifically include: First boundary conditions: The boundary condition at the tow cable's tip is a continuity condition, meaning the velocity at the tow cable's tip is the same as the velocity at the tow point on the underwater vehicle. Therefore, the boundary condition at the tip is... (19) In the formula, The translational speed of the underwater vehicle. The rotational angular velocity of the underwater vehicle. The velocity of the tow cable head in the tow cable local coordinate system.

[0044] Tail-end boundary conditions: When the tow cable trails the buoy system, the velocity of the tow cable trailing end is the same as the velocity of the buoy. The boundary condition for the tow cable trailing end is: (20) In the formula, The velocity of the trailing cable end in the local coordinate system of the trailing cable.

[0045] In one embodiment, in step S2, the motion control equations of the underwater vehicle and the buoy are propagated by time integration using the fourth-order Runge-Kutta method.

[0046] In one embodiment, in steps S1 and S3, the motion control equations of the tow cable are solved using the hybrid finite difference method. Specifically, the spatial derivative terms of the tow cable are discretized using a central difference scheme, transforming them into a system of ordinary differential equations about spatial discrete node variables. Subsequently, the system of ordinary differential equations is solved iteratively using an implicit time integration scheme.

[0047] In one embodiment, a time-domain dynamic simulation of an underwater towed rigid-flexible coupled system is performed based on a bidirectional closed-loop computational framework.

[0048] This embodiment uses a typical underwater towed system as an example to verify the effectiveness of the improved time-domain dynamics calculation method proposed in this invention. The system parameters are as follows: Underwater vehicle: Mass 500 kg, moment of inertia , , The hydrodynamic coefficients are obtained by CFD calculation; Towing cable: Total length 300 m, diameter 0.02 m, mass per unit length 1.5 kg / m, Young's modulus 5e10 Pa, normal drag coefficient 1.2, tangential drag coefficient 0.015; Buoy: Mass 80 kg, Displacement volume 0.78 m³, Moment of inertia , , The hydrodynamic coefficients under wave action are calculated using three-dimensional potential flow theory. Environmental conditions: Sea state 5, significant wave height 3.0 m, spectral peak period 8.0 s, irregular wave simulated using JONSWAP spectrum, surface current velocity 1.0 kn, direction consistent with wave direction; Simulation parameters: total duration 600 seconds, time step 0.02 s, the tow cable is uniformly divided into 200 micro-elements along the arc length.

[0049] Step 1: Establish a multi-coordinate system framework like Figure 2 As shown, establish the following coordinate system: Spatial fixed coordinate system A moving coordinate system fixed to the center of gravity of the underwater vehicle A local coordinate system used to describe the spatial shape and forces acting on the towing cable. and the following coordinate system fixed to the buoy's center of gravity. The transformation matrix between coordinate systems is shown in equation (18).

[0050] Step 2: Equations of independent motion for components The motion control equations for each component are shown in equations (1) to (15) in the previous embodiments. This embodiment directly adopts these equations and supplements the specific numerical solution details.

[0051] The motion control equations for underwater vehicles are: Equations (1) to (3) are rigid body dynamics equations. The external forces on the right end include: ideal fluid inertial force (additional mass force); negative buoyancy (gravity minus buoyancy); viscous drag (calculated from the drag coefficient provided by CFD); and towing cable force. and torque Provided by the tension at the beginning of the tow cable, the calculation is shown in step S1.

[0052] The control equations for the tow cable motion are: Equation (4) is a set of partial differential equations for the cable, containing six unknowns: three velocity components ( , , ) and three position coordinates ( , , However, equation (4) is actually in the form of velocity and tension as variables. To facilitate numerical solution, the cable is considered along the arc length. Discretization is performed using the mixed finite difference method: the spatial derivative is approximated by the central difference, resulting in a system of ordinary differential equations about time.

[0053] Buoy motion control equations: Equations (10) to (15) are the buoy motion equations. The external forces on the right end include gravity, buoyancy, wave force (calculated by potential flow theory), and the action of the tow cable tail end. and torque .

[0054] Step 3: Coupled Boundary Conditions The boundary condition at the tow cable's head is a continuity condition, meaning the velocity at the tow cable's head is the same as the velocity at the tow point on the underwater vehicle. Correspondingly, the velocity at the tow cable's tail is the same as the velocity of the buoy. The coupling conditions at the connection point are given by equations (19) and (20).

[0055] Step 4: Numerical Implementation of the Bidirectional Closed-Loop Computational Framework This step details the numerical operations of each substep to ensure strict timing for bidirectional data exchange.

[0056] S1: Calculation of Coupled Boundary Force Transfer (Current Moment) ) Known The form of the tow cable at any time (node ​​position) ) and velocity field (nodal velocities) The process involves solving the tow cable motion control equations to obtain the end-point tension. However, note that in the closed-loop framework, S1 actually calculates the end-point tension directly based on the known tow cable state (without resolving the tow cable, as the tow cable state is already obtained from S3 in the previous time step). A more accurate process is: at the beginning of each time step, the tow cable state is known (updated from the previous step S3). Therefore, S1 only needs to extract the end-point tension from the current tow cable state. Tension at discrete nodes (This can be obtained from strain calculations using the cable constitutive relation), thus yielding the form of the tension vector at the head end in the spatial frame. And calculate the moment about the center of gravity of the aircraft. ,in It needs to be switched to a space system.

[0057] Similarly, end tension and the torque on the buoy .

[0058] These forces and torques serve as inputs to S2.

[0059] S2: Rigid body state cooperative propulsion (from...) arrive ) Substituting the boundary forces obtained from S1 into the six-degree-of-freedom equations of the underwater vehicle and the buoy, and using the fourth-order Runge-Kutta method in parallel integration, we obtain... The rigid body state at any given moment.

[0060] Taking an aircraft as an example: Equations (1) to (3) are written in state-space form. Define the state vector. ,in For position and Euler angle, Let be the linear velocity and angular velocity in body coordinates. The dynamic equations can be written as:

[0061]

[0062] in, Here is the kinematic transformation matrix. This is the mass matrix (including additional mass). For other external forces. The steps of the fourth-order Runge-Kutta method are:

[0063]

[0064]

[0065]

[0066]

[0067] in, This is the state derivative. A fourth-order Runge-Kutta integral is also performed on the buoy; both are calculated independently and in parallel.

[0068] Points obtained The position of the spacecraft and speed and buoys , .

[0069] S3: Cable status synchronization update (from arrive ) Using the new position and velocity of the rigid body obtained in S2 as the imposed boundary conditions, the control equations for the towing cable motion are solved again, yielding... The cable shape and tension distribution at any given moment.

[0070] Discretization of the towing cable equations: using the hybrid finite difference method. Along the arc length Discretize the cable into Each microelement has nodes numbered from 1 to... In each internal node ( The spatial derivative is obtained by using central difference. Taking equation (4) as an example, its essence is a system of partial differential equations, which can be rewritten in terms of the time derivative. Here is the general formula of the discretized system of ordinary differential equations:

[0071] in, , It is obtained by central difference from equations (4)-(8).

[0072] Boundary conditions: Head ( ): Known Node velocity at time step Determined by the motion of the aircraft (Equation (19)). This condition is directly set as a first-type boundary condition. The velocity component in the equation.

[0073] Tail end ( Similarly, determined by the buoy's motion (Equation (20)), the following settings are made: The velocity component in the equation.

[0074] Time integration: Implicit Euler method is used to ensure numerical stability.

[0075]

[0076] Combining boundary conditions, form a conclusion about A system of nonlinear algebraic equations is proposed. A Newton-Raphson iterative approach is used to solve it, iterating until the residual norm is less than the tolerance. Each iteration requires calculating the Jacobian matrix. Solving for all nodes yields the results. Thus, the cable position (by integrating equation (9)) and tension distribution are obtained.

[0077] S4: System State Synchronization and Looping S2 obtained , Combined with the tow cable node states obtained in S3, a complete system state vector is formed. Then let Return to S1 to start the next time step, until the total simulation time is reached.

[0078] Note: In S1, the current state of the tow cable is known (i.e., obtained from the previous step S3), so S1 can directly extract the end tension without solving for the tow cable again. In this method, there is only one bidirectional transfer within each time step, and no sub-iterations are performed, but the consistency of coupling is guaranteed through implicit time integration.

[0079] In one embodiment, a rigid-flexible coupled two-way closed-loop time-domain dynamics calculation method for the underwater towing system is implemented based on a simulation system, wherein the simulation system includes: The model building unit constructs a rigid body six-degree-of-freedom motion model for an underwater vehicle, a rigid body six-degree-of-freedom motion model for a buoy, and a partial differential dynamics model for a towed cable continuous elastic body, and establishes coordinate system transformation relationships and physical parameter mappings between the models. The coupled boundary condition management unit defines the displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions between the underwater vehicle and the tow cable head end, and between the tow cable tail end and the buoy, as constraints for data exchange between the various models. The rigid body motion solver uses the fourth-order Runge-Kutta method to perform time integration on the six-degree-of-freedom motion ordinary differential equations of underwater vehicles and buoys, and outputs the position, attitude and velocity of the rigid body at the next moment. The flexible cable solver uses the hybrid finite difference method to spatially discretize and implicitly integrate the partial differential dynamic equations of the tow cable, and outputs the spatial position, velocity and tension distribution along the cable at each node. A bidirectional closed-loop coupled controller is connected to the rigid body motion solver, the flexible cable solver, and the coupled boundary condition management unit, and performs the following operations sequentially within each time step: The flexible cable solver is triggered to calculate the boundary tension and moment at the head and tail ends based on the current tow cable state; The boundary tension and torque are input as external loads to the rigid body motion solver, driving it to advance the rigid body state of the underwater vehicle and the buoy in parallel. The new position and velocity output by the rigid body solver are used as updated forced boundary conditions to trigger the flexible cable solver to synchronously correct the tow cable state. Collect the output results of the rigid body solver and the flexible cable solver to form the complete system state vector at the next moment; The system state memory stores the complete state sequence of the underwater vehicle, buoy, and tow cable generated by the bidirectional closed-loop coupled controller in time steps. The simulation output interface reads and outputs time-domain dynamic response data from the system state memory, including the motion trajectory and attitude history of the rigid body, the spatial configuration evolution of the tow cable, and the tension distribution history.

[0080] Reference Figure 3 The figure shows the time-varying tension curve at the tow cable's tip. The results of the proposed two-way closed-loop method are compared with those of the traditional sequential solution method (i.e., solving for the vehicle first, then the tow cable, and finally the buoy, without feedback). It can be seen that the proposed method yields a higher peak tension and a phase that better matches the wave excitation, while the sequential method exhibits significant phase lag and amplitude attenuation.

[0081] Reference Figure 4 As shown, the spatial configuration of the tow cable at different times demonstrates the dynamic deformation and wave propagation of the cable under wave action. These configurations are obtained by connecting the nodal coordinates, intuitively reflecting the motion characteristics of the system.

[0082] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towed system, characterized in that, The specific steps are as follows: Step 1: Establish a multi-coordinate system framework: Construct multiple coordinate systems to describe the spatial motion of the underwater vehicle, flexible tow cable, and surface buoy, including a global fixed coordinate system, a follower coordinate system fixed to the underwater vehicle, a follower coordinate system fixed to the buoy, and a local coordinate system to describe the local shape and force of the tow cable. Step 2: Establish independent motion control equations for components: Treat the underwater vehicle and buoy as rigid bodies and establish their six-degree-of-freedom motion equations respectively; treat the tow cable as a continuous elastic body and establish its dynamic partial differential equations, which include fluid resistance, inertial force, gravity, buoyancy and time-varying load terms of internal tension. Step 3: Construct an overall coupled dynamic model: By using displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions, the motion control equations of the underwater vehicle, tow cable, and buoy are coupled at the connection boundary to form an overall dynamic model that reflects the dynamic interaction between system components. Step 4: Construct a closed-loop computing framework for bidirectional data exchange for collaborative solving: Within each time step, execute the following sub-steps in strict timing order: S1. Coupled boundary force transmission: Based on the shape and velocity field of the tow cable at the current moment, solve the motion control equation of the tow cable to obtain the time-varying tension and torque acting on the underwater vehicle at the tow cable head end, and the time-varying tension and torque acting on the buoy at the tow cable tail end. S2. Rigid body state cooperative propulsion: The boundary forces and moments obtained in S1 are used as external loads and substituted into the six-degree-of-freedom motion equations of the underwater vehicle and the buoy respectively. The ordinary differential equations of each are integrated in parallel over time to predict the position, attitude and velocity of the underwater vehicle and the buoy at the next moment. S3. Synchronous Update of Tow Cable Status: The position and velocity of the underwater vehicle and buoy at the next moment predicted by S2 are used as the updated imposed boundary conditions. The motion control equation of the tow cable is solved again to obtain the overall shape, node velocity and tension distribution of the tow cable at the next moment that are consistent with the updated boundary. S4. System State Synchronization and Looping: Summarize the next-time states of all components obtained in S2 and S3, update the complete state vector of the entire coupled system, and proceed to the iterative calculation of the next time step; Step 5: Obtain the system dynamic response: Through the above closed-loop collaborative solution, obtain the complete dynamic response of the underwater towed system in the time domain, including the position, attitude, and velocity of the underwater vehicle and buoy, as well as the spatial configuration, tension distribution, and dynamic load history of the tow cable.

2. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In step two, the independent motion control equations for each component are as follows: The motion control equations for underwater vehicles are established based on the theorems of momentum and angular momentum, and their form is as follows: in, Let be the inertial matrix of the underwater vehicle in volume coordinates. The Coriolis force-centrifugal force matrix represents the force of an underwater vehicle moving in volume coordinates. and Let these be the generalized velocity vector and generalized acceleration vector of the underwater vehicle in the body coordinate system. It is the resultant force vector of all external forces and external moments acting on the underwater vehicle; The motion control equation for the towing cable is based on d'Alembert's principle, using the tension on the towing cable... Local velocity components , , and attitude angle , A system of partial differential equations established for the variables; Buoy motion control equations: adopting a six-degree-of-freedom rigid body motion equation similar to that of a ship, with external force terms including gravity, buoyancy, wave force, and tension and torque transmitted from the tail end of the tow cable.

3. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In step three, the coupling boundary conditions are specifically as follows: Bow boundary condition: The velocity at the bow of the tow cable is the same as the velocity at the tow point on the underwater vehicle, satisfying the condition. In the formula, The translational speed of the underwater vehicle. The rotational angular velocity of the underwater vehicle. This is the position vector of the tow cable tip in the underwater vehicle's follower coordinate system; This is the transformation matrix between the fixed spatial coordinate system and the underwater vehicle's moving coordinate system. The transformation matrix between the fixed spatial coordinate system and the local coordinate system of the towing cable. The velocity of the tow cable head end in the tow cable local coordinate system; Tail-end boundary condition: The velocity at the tail end of the tow cable is the same as the velocity at the tow point on the buoy, satisfying the condition. In the formula, Let be the buoy's translational speed. Let be the angular velocity of the buoy. is the position vector of the tow cable tail end in the buoy's following coordinate system; This is the transformation matrix between the buoy's moving coordinate system and the fixed spatial coordinate system. The transformation matrix between the fixed spatial coordinate system and the local coordinate system of the towing cable. The velocity of the trailing cable end in the local coordinate system of the trailing cable.

4. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In S2, the six-degree-of-freedom motion equations of the underwater vehicle and buoy are propagated by time integration using the fourth-order Runge-Kutta method, and the two are calculated in parallel.

5. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In S1 and S3, the control equations for the tow cable motion are solved using the hybrid finite difference method. Specifically, the spatial derivative terms of the tow cable are discretized using a central difference scheme, transforming the partial differential equations into a set of ordinary differential equations about spatial discrete node variables. Then, the set of ordinary differential equations is solved iteratively using an implicit time integration scheme.

6. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 5, characterized in that: The implicit time integral in the hybrid difference method employs the implicit Euler method or the Newton-Raphson iterative solution of the nonlinear algebraic equations to ensure numerical stability.

7. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In the closed-loop calculation framework of step four, the sequential process from S1 to S4 is executed only once within each time step, without internal sub-iterations. The consistency of the coupled state is achieved through the implicit time integration of the cable solver in S3.

8. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: The dynamic response obtained in step five includes: the six-degree-of-freedom motion history of the underwater vehicle and the buoy, the spatial position and velocity history of each node of the tow cable, the tension distribution history of the tow cable along the arc length, and the dynamic tension and torque history of the tow cable's head and tail ends.

9. The method for calculating the rigid-flexible coupled two-way closed-loop time-domain dynamics of an underwater towing system according to claim 1, characterized in that: In the multi-coordinate system framework, the transformation between the coordinate systems is achieved using the Euler angle direction cosine matrix. The transformation matrix between the servo coordinate system of the underwater vehicle and the buoy and the fixed spatial coordinate system is determined by the roll angle, pitch angle and yaw angle.

10. A rigid-flexible coupled bidirectional closed-loop time-domain dynamics simulation system for an underwater towed system, used to implement the rigid-flexible coupled bidirectional closed-loop time-domain dynamics calculation method for an underwater towed system as described in any one of claims 1-9; characterized in that, include: The model building unit constructs a rigid body six-degree-of-freedom motion model for an underwater vehicle, a rigid body six-degree-of-freedom motion model for a buoy, and a partial differential dynamics model for a towed cable continuous elastic body, and establishes coordinate system transformation relationships and physical parameter mappings between the models. The coupled boundary condition management unit defines the displacement coordination conditions, velocity continuity conditions, and force and torque balance conditions between the underwater vehicle and the tow cable head end, and between the tow cable tail end and the buoy, as constraints for data exchange between the various models. The rigid body motion solver uses the fourth-order Runge-Kutta method to perform time integration on the six-degree-of-freedom motion ordinary differential equations of underwater vehicles and buoys, and outputs the position, attitude and velocity of the rigid body at the next moment. The flexible cable solver uses the hybrid finite difference method to spatially discretize and implicitly integrate the partial differential dynamic equations of the tow cable, and outputs the spatial position, velocity and tension distribution along the cable at each node. A bidirectional closed-loop coupled controller is connected to the rigid body motion solver, the flexible cable solver, and the coupled boundary condition management unit, and performs the following operations sequentially within each time step: The flexible cable solver is triggered to calculate the boundary tension and moment at the head and tail ends based on the current tow cable state; The boundary tension and torque are input as external loads to the rigid body motion solver, driving it to advance the rigid body state of the underwater vehicle and the buoy in parallel. The new position and velocity output by the rigid body solver are used as updated forced boundary conditions to trigger the flexible cable solver to synchronously correct the tow cable state. Collect the output results of the rigid body solver and the flexible cable solver to form the complete system state vector at the next moment; The system state memory stores the complete state sequence of the underwater vehicle, buoy, and tow cable generated by the bidirectional closed-loop coupled controller in time steps. The simulation output interface reads and outputs time-domain dynamic response data from the system state memory, including the motion trajectory and attitude history of the rigid body, the spatial configuration evolution of the tow cable, and the tension distribution history.

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