Rigid-Flexible Coupling Simulation Method and System for Hull Section Lifting

Abstract

This invention relates to the field of lifting machinery dynamics, specifically a rigid-flexible coupling simulation method and system for lifting sections of a ship. It establishes a global coordinate system, uses Euler angles to describe the attitude of the lifted object, and abstracts the lifting system into a multi-stage force transmission structure. The entire lifting process is divided into three stages: connection, straightening, and motion. In the connection stage, the length of the lower rope section is fixed. In the straightening stage, the initial equilibrium configuration and tension are solved with the goal of minimizing the center of gravity height. In the motion stage, geometric solutions, relaxation detection, and tension solutions are performed iteratively. The force balance equations are integrated into a unified linear equation system, solved using Tikhonov regularized bounded least squares under non-negative constraints to achieve a more uniform tension distribution. Relaxed ropes are identified based on geometric criteria, and the active rope set is updated and the equation system is reconstructed through cascaded bidirectional propagation. After each solution step, a safety assessment index is output, and the attitude of the lifted object and load balance are verified. This solves the problems of static indeterminacy, geometric nonlinearity, and motion coupling superimposed on each other.

Description

Technical Field

[0001] This invention relates to the field of crane dynamics technology, specifically to a rigid-flexible coupling simulation method and system for hoisting ship sections. 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] The shipbuilding industry generally adopts a modular manufacturing system of "section construction and final assembly," and the lifting operations of large ship sections rely on gantry cranes. As the weight of sections continues to increase and the shape of the hull becomes increasingly irregular, the dual-crane lifting mode with two or more gantry cranes working together has become the norm in engineering projects.

[0004] In tandem lifting operations, the number of lifting lugs typically exceeds three. In this case, the number of independent constraints provided by the force balance equations is less than the number of unknown rope tensions, forming a typical static indeterminate problem. Theoretically, there are infinitely many sets of non-negative tension solutions that satisfy force balance, but the stress levels of each rope may vary significantly under different distribution schemes: the load may be concentrated on a few ropes, while the remaining ropes are nearly slack. In the worst-case scenario, some ropes will be severely overloaded, posing a risk of breakage and directly threatening lifting safety.

[0005] Existing hoisting simulation tools are mainly designed for single-machine, single-point hoisting or simple multi-point hoisting, lacking systematic mathematical means for handling static indeterminate problems in multi-machine collaborative scenarios. Some tools approximate solutions through preset weight allocation or simplified equivalent methods, but the weight setting relies on experience and lacks mechanical rigor, making it difficult to guarantee physically consistent and reasonably balanced solutions under arbitrary configurations. When the lifting lug layout is asymmetrical, the center of gravity of the hoisted object is off-center, or the position and posture change during operation, the accuracy of simplified methods further decreases, failing to provide reliable quantitative basis for engineering safety assessment. Summary of the Invention

[0006] This invention provides a rigid-flexible coupling simulation method and system for hoisting ship hull sections. In multi-lumen configurations where multiple gantry cranes work together to hoist ship hull sections, the undetermined force balance equations lead to static indeterminacy of the system, resulting in infinitely many sets of non-negative tension feasible solutions. Existing methods lack a systematic mathematical means to automatically select those that satisfy force balance and tend to distribute the load evenly, making it difficult to avoid the risk of severe overloading of individual ropes. This invention provides a reliable quantitative basis for engineering safety assessment.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] The first aspect of this invention provides a rigid-flexible coupling simulation method for the hoisting of ship hull sections, comprising the following steps:

[0009] Step S1: Establish a global rectangular coordinate system, use Euler angles to describe the spatial attitude of the suspended object, use the body coordinate system to parameterize the global coordinates of each lifting lug, and abstract the hoisting system into a multi-level force transmission structure of "trolley-upper rope-lifting block node-lower rope-lifting lug-suspended object".

[0010] Step S2: Divide the entire hoisting process into three stages: connection stage, straightening stage, and motion stage. Each stage has different constraint structures and solution tasks.

[0011] During the connection phase, the spatial distance between the two ends of each lower rope segment is recorded as its physical length and fixed while the rope is in a slack state, without performing mechanical solutions.

[0012] During the straightening stage, with the goal of minimizing the height of the center of gravity of the suspended object, the initial equilibrium configuration and initial rope tension of the system are solved by nonlinear constraint optimization under the constraints of the lower rope length equality and the upper rope length inequality.

[0013] During the motion phase, as the control variables are gradually updated, geometric solutions, relaxation detection, and tension solutions are cyclically executed.

[0014] Step S3: In each solution step, the force balance equation of the suspended object and the force balance equation of each suspension node are integrated into a unified linear equation system about the tension of all active ropes. The Tikhonov regularized bounded least squares method is used to solve the equations under non-negative constraints, so that the solution results tend to uniform load distribution under the premise of satisfying force balance.

[0015] Step S4: Identify slack ropes based on geometric criteria, update the active rope set through cascading bidirectional propagation between the upper rope, the hoisting node and the lower rope, reconstruct the equation set after removing the geometric constraints and tension equation terms corresponding to the slack ropes, and iterate alternately through slack detection and geometric solution until the active rope set converges.

[0016] Step S5: After each solution step is completed, output safety assessment indicators for the active rope and comprehensively verify the attitude of the suspended object and the load balance of each machine.

[0017] Furthermore, the global rectangular coordinate system takes the ground reference point of the main assembly platform as the origin, with the X-axis along the direction of the gantry crane main beam, the Y-axis along the direction of the gantry crane track, and the Z-axis vertically upward;

[0018] The body coordinate system takes the center of gravity of the suspended object as its origin, and the coordinates of each lifting lug in the body coordinate system are calculated and fixed when the connection stage is completed.

[0019] The global coordinates of the i-th lug at any given time are calculated by the following formula:

[0020] ;

[0021] in For the global coordinates of the center of gravity of the suspended object, Let ZYX be the order Euler angle rotation matrix. Let be the fixed coordinates of the i-th lug in the body coordinate system; the Euler angles are limited to the range of values ​​within . Inside.

[0022] Furthermore, the rope length fixing process during the connection phase is as follows: the connection operation is performed when the rope is fully slack and the hoisting platform is vertically suspended directly below the trolley. The initial coordinates of the hoisting platform nodes are determined by offsetting the trolley coordinates downwards by the length of the upper rope segment. The physical length of each lower rope segment is fixed according to the spatial distance between the two endpoints, as shown in the following formula:

[0023] ;

[0024] in, Let be the fixed physical length of the i-th lower rope segment; For the connection phase The initial global coordinates of the first hoist lug; Let j(i) be the initial coordinates of the j(i)th hoisting node connected to the i-th lower rope segment; j(i) represents the hoisting number to which the i-th lower rope segment belongs. This represents the total number of rope segments; during the connection phase, only the rope connection relationships and physical lengths are recorded, and tension calculation is not performed.

[0025] Furthermore, with the objective of minimizing the Z-coordinate of the suspended load's center of gravity, a nonlinear constrained optimization problem is constructed under the constraints of equality of the lower rope length and inequality of the upper rope length. This problem is solved iteratively using a sequential quadratic programming method. The initial conjecture is generated based on the simplified assumption that "all ropes are vertical," and the estimated height of the suspended load's center of gravity is:

[0026] ;

[0027] in, This represents the initial estimate of the height of the center of gravity of the suspended load. Indicates the number of active trolleys involved in the hoisting operation; This represents the height of the main beam of the j-th car; This represents the initial length of the j-th upper rope segment set during the connection phase; Indicates the total number of rope segments in the lower section; This represents the physical length of the i-th lower rope segment after solidification during the connection stage. The above estimate is only used as an initial guess for constructing the nonlinear constraint optimization during the straightening stage. The actual coordinates of the center of gravity of the suspended object are determined by the subsequent lower rope segment length equality constraint, the upper rope segment length inequality constraint, and the geometric optimization solution.

[0028] After the geometric solution converges, the unit direction vectors of each rope are substituted into the tension equations, and the initial tension is solved using the Tikhonov regularized bounded least squares method.

[0029] Furthermore, the solution strategy for the tension equations is as follows: the force equilibrium of the suspended object provides 6 equations. Each active hoisting node provides 3 force balance equations, which together constitute... A system of linear equations is solved using bounded least squares (with a lower bound of zero) to ensure non-negativity of tension. When the system is statically indeterminate, Tikhonov regularization is introduced, with the regularization coefficient and the expected uniform tension of each rope used as the basis for the solution. The solution guides the load distribution towards uniformity. When each hoisting bar is connected to only a single lug, the hoisting bar node equations degenerate into three-point collinear constraints, which is completely consistent with the solution strategy for multi-lug configurations.

[0030] Furthermore, the cascaded bidirectional relaxation propagation and constraint reconstruction process is as follows:

[0031] After the geometric solution converges, if the distance from the hoisting node to the trolley is less than the set length of the upper rope and exceeds the preset tolerance, the upper rope is determined to be slack and the tension is set to zero.

[0032] If the distance between the two ends of a lower section of rope deviates from its physical length by more than the preset tolerance, the rope is determined to be slack and the tension is set to zero.

[0033] When the upper section of the rope becomes slack, all the lower sections of the rope connected to the hoisting machine also become slack (propagation downwards); when all the lower sections of the rope of a hoisting machine become slack, the upper section of the rope of the hoisting machine is also determined to be slack (propagation upwards).

[0034] By removing the geometric constraints and tension equations corresponding to the slack rope, the equations are reconstructed and solved again based on the remaining active rope.

[0035] Furthermore, the geometric tolerance of relaxation detection is as follows: the relaxation tolerance of the upper rope segment and the relaxation tolerance of the lower rope segment are both set values; when the number of iterations of relaxation detection and geometric solution exceeds the upper limit, a convergence warning is output, and the convergence state of the active rope set is used as one of the judgment conditions of the adaptive re-solution mechanism.

[0036] Furthermore, the single-step solution process for the motion phase includes the following steps:

[0037] Step S201: Using the equilibrium configuration from the previous step as the initial guess for hot start, perform geometric solution and relaxation detection iterations until the active rope set converges;

[0038] Step S202: Based on the converged set of active ropes, recalculate the unit direction vectors of all active ropes and construct the coefficient matrix A and the right-hand vector b of the tension linear equation system.

[0039] Step S203: Apply Tikhonov regularization bounded least squares solution to obtain all active rope tensions, and relax the rope tensions to strictly zero.

[0040] Step S204: Calculate the force balance residuals and verify the non-negativity constraints;

[0041] Step S205 establishes the new equilibrium configuration and rope tension as the current system state for use in the next operation step of hot start-up and safety assessment.

[0042] Furthermore, when the force balance residual, torque residual, or active rope set convergence state does not meet the preset conditions, an adaptive re-solution mechanism is activated, including reducing the step size, using the successful convergence step as a hot start for re-solution, increasing the local iteration upper limit, or regenerating the initial guess of the hoisting node. If multiple re-solutions still do not meet the conditions, an infeasibility alarm is output.

[0043] A second aspect of the present invention provides a system for implementing the above-described method, comprising:

[0044] The coordinate system and pose modeling unit is configured to: establish a global rectangular coordinate system, use Euler angles to describe the spatial attitude of the suspended object, and use the body coordinate system to parameterize the global coordinates of each lug;

[0045] The three-stage process management unit is configured to divide the entire hoisting process into three stages: connection stage, straightening stage, and motion stage. Each stage has different constraint structures and solution tasks.

[0046] The solution unit is configured to integrate the force balance equations of the suspended object and the force balance equations of each suspension node into a unified linear equation system with respect to the tension of all active ropes in each solution step. The Tikhonov regularized bounded least squares method is used to solve the equations under non-negative constraints, so that the solution results tend to uniform load distribution under the premise of satisfying force balance.

[0047] The relaxation detection and constraint reconstruction unit is configured to: identify relaxed ropes based on geometric criteria, update the active rope set through cascaded bidirectional propagation between the upper rope, the hoisting node and the lower rope, reconstruct the equation set after removing the geometric constraints and tension equations corresponding to the relaxed ropes, and iterate alternately through relaxation detection and geometric solution until the active rope set converges.

[0048] The evaluation and verification unit is configured to output safety evaluation indicators for the active rope after each solution step and to comprehensively verify the attitude of the suspended object and the load balance of each machine.

[0049] Compared with existing technologies, one or more of the above technical solutions have the following beneficial effects:

[0050] 1. The entire process of hoisting the ship sections is divided into three stages: connection, straightening, and motion. Each stage corresponds to different constraint structures and solution tasks. This avoids problems such as excessive constraint application, false initial tension, and unstable motion state switching caused by mixing relaxed connection, initial tension, and motion update into the same set of equations. This ensures that the change of constraint state during the hoisting process is consistent with the actual physical process, laying a geometric foundation of physical autonomy for subsequent mechanical solutions.

[0051] 2. The force balance equations of the hoisting object and the force balance equations of each hoisting node are integrated into a unified linear equation system. The Tikhonov regularized bounded least squares method is used to solve this system under non-negative constraints, systematically solving the static indeterminacy problem in coordinated hoisting with multiple lifting lugs and cranes. The regularization term guides the solution results to tend towards uniform load distribution while satisfying strict force balance, effectively avoiding false negative tension, extreme off-center loading, or unnecessary local rope overload, making the tension solution both physically reasonable and engineering applicable.

[0052] 3. Through a relaxation identification and cascaded bidirectional propagation mechanism based on geometric criteria, the relaxation state is automatically transmitted between the upper rope, the suspension node, and the lower rope, and the active rope set and equation system are reconstructed accordingly. This mechanism ensures the physical autonomy of the rope's transition from tension to relaxation and from relaxation to retension during motion, avoiding non-physical situations such as "the upper rope is relaxed while the lower rope is still incorrectly constrained" or "the lower rope is completely relaxed while the upper rope still retains false tension" that may occur with unidirectional propagation, significantly improving the numerical stability of constraint switching during motion.

[0053] 4. After each solution step, safety assessment indicators such as the utilization rate of the active rope output and the safety factor are evaluated, and the attitude of the suspended object and the load balance of each machine are comprehensively checked. The simulation results are directly converted into an actionable adjustment basis for engineering. The closed-loop feedback between simulation solution and safety assessment can serve rope selection, optimization of lifting point layout, review of operation plan and modification of pre-operation contingency plan, and has good numerical robustness and engineering practical value. Attached Figure Description

[0054] 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.

[0055] Figure 1 This is a schematic diagram of the coordinate system of a hoisting system provided in one or more embodiments of the present invention;

[0056] Figure 2 A schematic diagram of a gantry crane structure provided for one or more embodiments of the present invention;

[0057] Figure 3This is a schematic diagram of the connection and straightening stages provided in one or more embodiments of the present invention, comparing two states: rope slack (connection stage) and rope taut (straightening stage);

[0058] Figure 4 The diagrams provided in one or more embodiments of the present invention illustrate the effects of trolley movement and rope length adjustment on the system's equilibrium configuration before and after motion phase control.

[0059] Figure 5 A schematic diagram of a working condition simulation model provided for one or more embodiments of the present invention;

[0060] Figure 6 The overall flowchart of the three-stage simulation framework provided for one or more embodiments of the present invention. Detailed Implementation

[0061] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0062] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration 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.

[0063] The reason why existing technologies struggle to effectively handle statically indeterminate problems is that traditional solution approaches treat tension distribution problems as purely mathematical optimization problems, without introducing guiding criteria with clear engineering physical significance. Among infinitely many feasible solutions, without additional distribution preference information, the solver may output solutions with extreme off-center loading, or even generate physically meaningless negative tensions, rendering the calculation results unsuitable for direct engineering safety assessments.

[0064] In addition to static indeterminacy, collaborative lifting scenarios involving two gantry cranes present two other types of mechanical challenges: first, a strongly nonlinear relationship exists between rope tension and the orientation of the load, i.e., a geometric nonlinearity problem; second, the two cranes are mechanically coupled through the load, and any operation will transmit influence to each other, i.e., a motion coupling problem. However, the root cause of both of these problems lies in the unresolved static indeterminacy. If a physically consistent tension solution cannot be obtained, the coupled iteration of geometric configuration and mechanical equilibrium will lose its convergence basis, and the load transfer between the two cranes cannot be accurately described.

[0065] Therefore, this scheme proposes a rigid-flexible coupling simulation method and system for the hoisting of ship sections. It elevates the tension solution from a purely mathematical problem to an optimization problem with engineering load distribution criteria. By introducing a Tikhonov regularization term into the non-negative constrained bounded least squares framework, and taking the tendency of the load to be uniformly distributed as the guiding direction, it realizes the automatic selection of a unique solution that satisfies force balance and meets engineering expectations from infinitely many feasible solutions. Furthermore, it decouples and iterates the geometric solution and the mechanical solution under a unified framework, and finally systematically solves the challenge of multiple mechanical difficulties superimposed on each other.

[0066] like Figure 6 As shown, the rigid-flexible coupling simulation method for hoisting ship sections includes the following steps:

[0067] Step 1: Establish a coordinate system.

[0068] Establish a global rectangular coordinate system O-XYZ, and use the ZYX sequence Euler angles (yaw angles). Pitch angle Roll angle Describe the spatial attitude of the suspended object; establish a body coordinate system with the center of gravity of the suspended object as the origin, permanently fix the local coordinates of each lifting lug when the connection stage is completed, and calculate the global coordinates of the lifting lugs at any subsequent time by the following formula:

[0069] ;

[0070] The hoisting system is abstracted as a multi-stage force transmission structure of "trolley - upper rope - hoisting panel node - lower rope - lifting lug - hoisted object", and the hoisting panel is equivalent to a massless force convergence node.

[0071] Step Two: Three-Stage Process Management.

[0072] The entire hoisting process is modeled in the order of connection, straightening, and motion phases. In the connection phase, the physical lengths of all lower rope segments are fixed in the slack state, and no mechanical solutions are performed. In the straightening phase, the initial equilibrium configuration and tension of the system are solved with the goal of minimizing the height of the center of gravity of the hoisted object. In the motion phase, geometric solutions, slack detection, and tension solutions are performed cyclically as the control variables are updated.

[0073] Step 3: Solve the geometric constraints.

[0074] Given the rope length constraint and the trolley position, construct a structure based on the height of the center of gravity of the suspended object. The nonlinear constrained optimization problem with the objective of minimization, under the conditions of satisfying the equality constraint of the lower rope length and the inequality constraint of the upper rope length, uses the Sequential Quadratic Programming (SLSQP) method to iteratively determine the pose of the suspended object. ) and coordinates of each hoisting node The solution at the previous time step is used as the initial guess for hot start to ensure the convergence stability of multi-step iteration.

[0075] Step 4: Solve the unified tension equations.

[0076] The equivalent total gravity is calculated using the inertia coefficient method. Substitute the force balance equations of the hoisting object (6 equations) and the force balance equations of each hoisting row node (3 equations per hoisting row) into a unified linear equation system. Tikhonov regularized bounded least squares (non-negativity constraint) is used. The regularization formula is as follows. Solving under non-negativity constraints guides the results towards uniform load distribution.

[0077] .

[0078] Step 5: Relaxation detection and constraint reconstruction.

[0079] Slack ropes are identified based on geometric criteria (5cm tolerance for the upper rope and 1cm tolerance for the lower rope), and cascaded bidirectional propagation is performed (slack in the upper rope corresponds to slack in all lower ropes; slack in all lower ropes corresponds to slack in the upper rope). The set of active ropes is updated based on the propagation results, and the geometric constraints and tension equation terms corresponding to the slack ropes are removed before reconstructing the equation set. Slack detection and geometric solution are iterated alternately until the set converges, with an upper limit of 5 iterations.

[0080] Step Six: Safety Assessment and Results Output.

[0081] After each solution step, output utilization for all active ropes. and safety factor The system comprehensively verifies the deflection of the suspended object's attitude, the height range of the hoisting nodes, and the load balance of the two machines. When the evaluation results show that the rope utilization rate is too high, the safety factor is insufficient, the suspended object's attitude exceeds the limit, or the load ratio of the two machines is unbalanced, corresponding adjustment suggestions are generated, and the adjusted scheme is re-input into the simulation solution process to form an engineering optimization closed loop of "simulation solution - safety assessment - adjustment suggestions - re-solution".

[0082] The initial guess construction rule for geometric solution is as follows: the estimated height of the center of gravity of the suspended object is the average height of each trolley minus the average length of the upper rope and then the average length of the lower rope; the initial height of the lifting platform node is the trolley height minus 0.95 times the length of the upper rope (slightly lower than the tautness critical value), ensuring that the initial state is within the constraints and avoiding misjudgment of constraint activity. During the motion phase, the increment of single-step operation should be limited within a reasonable range to ensure the convergence stability of geometric optimization. For specific limits, please refer to claim 9.

[0083] The collaborative configuration of double gantry cranes is not fundamentally different from that of single-mechanism configurations at the level of mechanical equations. All ropes of the two cranes are uniformly included in the same set of equations for solution. However, at the level of engineering operation, attention needs to be paid to three special characteristics: yaw-rotation coupling (rotation caused by differential operation), unbalanced load distribution (determined by the relative position of the lifting lugs and the center of gravity), and the requirement for synchronous operation accuracy (tilting caused by inconsistent rope speeds).

[0084] when When it is a single-lug direct connection configuration, the force balance equation of the suspension node automatically degenerates into the collinear constraint of the upper and lower ropes, and the tension values ​​of the upper and lower ropes are equal. No additional branch processing logic is needed, and it is completely consistent with the solution framework of the multi-lug configuration.

[0085] The detailed process of this solution is described below with reference to the accompanying drawings.

[0086] System physical modeling:

[0087] like Figure 1 As shown, this scheme uses a global rectangular coordinate system O-XYZ as a unified reference benchmark. The origin is taken as the ground reference point of the assembly platform (usually the ground projection of the intersection of the center lines of the two gantry crane tracks). The X-axis is along the direction of the gantry crane main beam, the Y-axis is along the direction of the gantry crane track, and the Z-axis is vertically upward. It should be noted that the specific directions of the X and Y axes depend on the installation orientation of the gantry crane on site, and the model frame is not affected.

[0088] The system's hierarchical force transmission structure is "trolley - upper rope - hoisting platform node - lower rope - lifting lug - suspended object". The gantry crane structure and trolley are considered rigid bodies, the wire rope is considered a flexible straight section subjected only to tension, the hoisting platform is equivalent to a massless force convergence node, and the suspended object is a spatial rigid body. This treatment allows the "rigid trolley / suspended object" and the "flexible rope" to be solved simultaneously within the same mechanical framework, reflecting the rigid-flexible coupling nature of the system. The spatial attitude of the suspended object is described using ZYX sequential Euler angles, which are, in order, yaw angles around the Z-axis. Tilting angle around the Y-axis Roll angle around the X-axis The range of values ​​for the three Euler angles is limited to... It is suitable for normal hoisting scenarios where large-angle overturning does not occur.

[0089] A local coordinate system is established with the center of gravity of the suspended object as the origin. The coordinates of each lifting lug in this local coordinate system are calculated and permanently fixed when the connection phase is completed. Thereafter, at any given time, the global coordinates of the i-th lifting lug are determined by the current center of gravity coordinates of the suspended object and the rotation matrix.

[0090] ;

[0091] in, Let i be the global coordinates of the i-th lug. For the global coordinates of the center of gravity of the suspended object, Let ZYX be the order Euler angle rotation matrix. Let be the fixed coordinates of the i-th lug in the body coordinate system; the Euler angles are limited to the range of values ​​within . Inside.

[0092] The above formula is the calculation formula for the coordinates of the lifting lug, which will be used repeatedly in the constraint equations and mechanical equations at all stages.

[0093] By adopting the assumption of an inelastic rope and ignoring the elastic elongation of the wire rope, the geometric configuration is independently determined by the rope length constraint equation, without depending on the tension value. This achieves the two-step decoupling of geometric and mechanical solutions, which is the key physical basis for the step-by-step solution strategy of this scheme.

[0094] Figure 2 This paper illustrates the hierarchical topology of components in a dual-gantry crane cooperative configuration. In this configuration, the two gantry cranes are located on different tracks, and their respective trolleys are connected to the same load, sharing the entire weight of the load. The two cranes are strongly mechanically coupled through the force balance equations of the load, making independent analysis impossible. A unified global equation set must be established to solve all unknowns at once. At the mechanical equation level, the dual-gantry crane cooperative configuration is not fundamentally different from the single-mechanism configuration; all ropes of the two cranes are uniformly incorporated into the same equation set for solution, requiring no additional coordination.

[0095] During actual hoisting, the load is in a state of accelerated motion and exhibits inertial force. This scheme uses the inertia coefficient method to convert the dynamic load into an equivalent static load: equivalent total gravity. Where k is the inertia coefficient ( (The default value is 1.1), where m is the mass of the suspended object. The value of k reflects the degree of influence of vertical acceleration during hoisting and should be selected according to the specific working conditions and relevant specifications. This treatment transforms the problem into a statics problem at each calculation moment, significantly reducing the computational complexity. It is suitable for low-speed, gradual operation of gantry cranes on land assembly platforms.

[0096] The above physical model is based on the following engineering simplification assumptions, and users should apply this solution within its applicable scope:

[0097] (1) Rope straight line assumption: Ignore the self-weight of the wire rope and regard each section of the rope as a straight line segment only subjected to tension. This is applicable to engineering scenarios where the rope diameter is negligible relative to the rope length.

[0098] (2) Massless Lifting Mesh Assumption: The mass of the lifting mesh is relatively small compared to the mass of the suspended object, and the influence caused by its mass is ignored within the engineering accuracy range. In this method, the lifting mesh is not treated as a rigid body with independent dynamic equations, but is equivalent to a massless force convergence node between the upper and lower ropes. For a lifting mesh with at least one active connecting rope, its node coordinates are determined by the corresponding rope length geometric constraints and force balance relationships. For a lifting mesh where the upper rope and all the lower ropes it connects to are judged to be relaxed, the lifting mesh node is no longer used as a geometric unknown in the equation system construction. Its coordinates are only retained as state records. The display coordinates can be given by using the position of the previous convergence step or by vertical projection of the current trolley position. These coordinates are not included in the suspended object pose solution, tension solution, and safety check. When the corresponding rope meets the tension candidate conditions again in subsequent operations, the lifting mesh node and its related ropes are reintroduced into the candidate constraint set.

[0099] (3) Quasi-static assumption: The dynamic load is equivalent to the static load by using the inertia coefficient method, and the lateral disturbance forces such as wind load and impact are ignored. It is suitable for low-speed and gradual operation scenarios of land assembly platform. For high dynamic or strong lateral disturbance conditions, the value of the inertia coefficient needs to be amplified accordingly.

[0100] (4) Rigid body lifting assumption: Ignore the elastic deformation of the hull sections during the lifting process. This is applicable to segmented structures with high stiffness and whose elastic deformation is negligible relative to the lifting displacement.

[0101] (5) Inelastic rope assumption: The elastic elongation of the wire rope is ignored, and the rope length is regarded as a rigid geometric constraint. This assumption is also the premise for the decoupling of the two steps of geometric solution and mechanical solution. For scenarios that need to consider dynamic transient response, this assumption needs to be relaxed.

[0102] Three-stage process management:

[0103] From a physical perspective, hoisting operations involve two fundamental transitions in constraint states from preparation to completion, naturally dividing them into three distinct phases: the connection phase, the straightening phase, and the movement phase. The transition between these three phases is actively triggered by the operator: "Connection → Straightening" is triggered by confirming the connection is complete and preparing for hoisting; "Straightening → Movement" is triggered by issuing the first movement or rope retraction command.

[0104] During the connection phase, the connection operation is performed when the ropes are fully slack and the hoisting platform is vertically suspended directly below the trolley. At this point, the initial coordinates of the hoisting platform nodes are determined by offsetting the trolley coordinates downwards by the length of the upper rope segment, and the upper rope segment is in a critical state of vertical tension. For each lower rope segment, the spatial distance between its two endpoints at the moment of connection is permanently fixed as the physical length of that rope. The fixed length of the lower rope segment remains constant throughout the simulation and is the sole geometric input for all subsequent phases from the connection phase. The physical characteristic of the connection phase is that the suspended object is not yet under stress, and no mechanical solutions are performed. Once the rope length is fixed, the system can proceed to the straightening phase.

[0105] During the straightening phase, the transition from slack to tension in actual hoisting is a dynamic process. This scheme adopts a quasi-static equivalent: the final state of the straightening process is directly defined as the state where all ropes are fully straightened and the system is in initial static equilibrium. The physical basis for this equivalence is that the final state of the straightening process is precisely the equilibrium configuration with the lowest potential energy of the system, which is completely consistent with the extreme value solution of the objective function obtained by geometric solution. Therefore, the initial equilibrium configuration can be directly obtained by solving the extreme value problem without simulating the dynamic transition process. After straightening, compared with the connection phase, the hoisting nodes shift from a vertical position to a new equilibrium position, tilting towards the hoisting lugs they are connected to. The upper section of the rope changes from vertical to tilted, and the suspended load descends to the position with the lowest potential energy.

[0106] During the motion phase, the operator drives the system to transition from one equilibrium state to the next by adjusting two types of control variables: first, position movement, which involves adjusting the coordinates of a gantry crane's trolley along the main beam direction, or the coordinates of the entire gantry crane along the track direction; second, rope length adjustment, which involves changing the length of the upper rope by raising or lowering the winch (positive values ​​for lowering the rope, negative values ​​for raising the rope), while the length of the lower rope remains constant throughout the process. After each adjustment, the system recalculates the new static equilibrium configuration and rope tension until the operation is completed.

[0107] Specifically:

[0108] The rope length fixing process during the connection phase is as follows: The connection operation is carried out when the rope is completely slack and the hoist is vertically suspended directly below the trolley. The initial coordinates of the hoist node are determined by offsetting the trolley coordinates downward by the upper rope length.

[0109] The physical length of each lower section of the rope is fixed according to the spatial distance between its two ends:

[0110] ;

[0111] in, Let be the fixed physical length of the i-th lower rope segment; For the connection phase The initial global coordinates of the first hoist lug; Let j(i) be the initial coordinates of the j(i)th hoisting node connected to the i-th lower rope segment; j(i) represents the hoisting number to which the i-th lower rope segment belongs. This represents the total number of rope segments. During the connection phase, only the rope connection relationships and physical lengths are recorded; tension calculation is not performed.

[0112] The initial equilibrium solution process during the straightening stage is as follows: With the goal of minimizing the Z-coordinate of the center of gravity of the suspended object, a nonlinear constraint optimization problem is formed under the constraints of the lower rope length equality and the upper rope length inequality. The problem is solved iteratively using the Sequential Quadratic Programming (SLSQP) method.

[0113] The initial guess was generated based on the simplified assumption that "all ropes are vertical". The height of the center of gravity of the suspended load was estimated as the average height of each trolley minus the average length of the upper rope and then the average length of the lower rope.

[0114] ;

[0115] in, This represents the initial estimate of the height of the center of gravity of the suspended load. Indicates the number of active trolleys involved in the hoisting operation; This represents the height of the main beam of the j-th car; This represents the initial length of the j-th upper rope segment set during the connection phase; Indicates the total number of rope segments in the lower section; This represents the physical length of the i-th lower rope segment after solidification during the connection stage. The above estimate is only used as an initial guess for constructing the nonlinear constraint optimization during the straightening stage. The actual coordinates of the center of gravity of the suspended object are determined by the subsequent lower rope segment length equality constraint, the upper rope segment length inequality constraint, and the geometric optimization solution.

[0116] After the geometric solution converges, the unit direction vectors of each rope are substituted into the tension equations, and the initial tension is solved using the Tikhonov regularized bounded least squares method.

[0117] Geometric constraint solution:

[0118] The goal of the geometric solution is: given all lower rope lengths (fixed values), the current coordinates of each trolley, and the set value of the upper rope length, to solve for the position of the hoisting nodes and the orientation of the suspended load (center of gravity coordinates and three Euler angles), totaling... An unknown quantity.

[0119] Construct a structure that minimizes the Z-coordinate of the center of gravity of the suspended load. This is a nonlinear constrained optimization problem with the objective function : The constraints are: equality constraint of lower rope length (the distance between the two ends of each lower rope segment equals its fixed physical length) and inequality constraint of upper rope length (the distance from the hoisting node to the trolley does not exceed the set length of the upper rope segment; equality indicates tension). The problem is solved iteratively using the Sequential Quadratic Programming (SLSQP) method. Objective function The physical meaning is: among all geometric configurations that satisfy the rope length constraint, the suspended object naturally falls to the position with the lowest potential energy, which is the equilibrium state that the system automatically reaches when the rope is straightened under the action of gravity.

[0120] The quality of solving nonlinear optimization problems largely depends on the quality of the initial guess. The initial estimate of the center of gravity height of the suspended object is based on the simplified assumption that "all ropes are vertical," taking the average height of each trolley minus the average length of the upper rope and then the average length of the lower rope. The initial height of the hoisting node is the trolley height minus 0.95 times the length of the upper rope, slightly lower than the position when the upper rope is taut. The reason for using a coefficient of 0.95 instead of 1.0 is that if the initial guess happens to be on the constraint boundary, the numerical calculation is prone to misjudgment of constraint activity, affecting the stability of the first iteration; a slight decrease of 5% ensures that the initial state is clearly inside the constraints, ensuring that the optimizer starts from within the feasible region, improving the reliability of the first iteration.

[0121] During the motion phase, the equilibrium configuration of the previous operation step is used as the initial guess for hot start to ensure the convergence stability of multi-step iterations. When the step size is too large, the configuration difference between two adjacent steps exceeds the linear approximation range, the initial guess deviates too much from the true solution, and geometric optimization may converge to an incorrect local extremum. Therefore, it is recommended that the single-step movement increment not exceed 0.5 m and the single-step rope extension / retraction amount not exceed 0.5 m. When a large adjustment is required, it should be performed in multiple steps.

[0122] Solving the unified tension equations:

[0123] After determining the positions of each node through geometric solutions, mechanical equations are established. Since the mass of the hoisting gantry is zero, the resultant force of all rope tensions acting on the nodes of the gantry must be strictly zero; if the resultant force is not zero, the zero-mass node will generate infinite acceleration, violating physical reality. Therefore, the upward tension of the upper ropes at each node of the gantry is balanced by the resultant force of all lower ropes downwards, providing three linear equations for each active gantry.

[0124] The solution strategy for the tension equations is as follows:

[0125] The force balance of the suspended object provides 6 equations. Each active hoisting node provides 3 force balance equations, which together constitute... A system of linear equations;

[0126] First, bounded least squares (with a lower bound of zero) is used to ensure that the tension is non-negative. Then, when the system is statically indeterminate, Tikhonov regularization is further introduced, using the regularization coefficient λ and the expected uniform tension of each rope. The solution is guided towards uniform load distribution;

[0127] The physical meaning of the regularization term is to guide the rope forces to tend to be evenly distributed when there is no additional constraint information, which conforms to the principle of maximum entropy. When each hoisting bar is connected to only a single lug, the hoisting bar node equation automatically degenerates into a three-point collinear constraint, which is completely consistent with the solution strategy for multi-lug configurations and does not require the introduction of additional branch processing logic.

[0128] Establish the force equilibrium equation for the entire suspended load: the sum of the vertical components of the tension in all active lower rope segments equals the equivalent total weight of the suspended load. The horizontal component automatically balances to zero; the moment balance equation with the center of gravity as the moment center gives one equation each around the X, Y, and Z axes, for a total of 6 equations. The upper section of the rope does not directly act on the suspended object, but only indirectly affects the tension of the lower section of the rope through the force balance at the lifting nodes.

[0129] The force balance of the suspended object (6 equations) and the force balance of each active hoist node (3 equations per hoist) are integrated into a unified linear equation system concerning the tension of all active ropes. All rope tensions (including the upper and lower rope sections) are solved at once within the same set of equations, eliminating the need for additional coordination between the two gantry cranes. This is a key method for handling motion coupling.

[0130] First, a bounded least squares method (with a lower bound of zero) is used to solve the problem, strictly ensuring that the tension is non-negative and avoiding solutions with negative tension that have no physical meaning. When the system is statically indeterminate (which usually occurs when there are more than three independent equations), Tikhonov regularization is further introduced, adding a regularization term to the objective function with a regularization coefficient. and the expected uniform tension of each rope The method guides the solution to a minimum norm solution that tends towards uniform load distribution. The physical meaning of the regularization term is that, without additional constraint information, the rope forces tend to be uniformly distributed, conforming to the maximum entropy principle. Compared to unconstrained least squares, this method effectively avoids spurious negative tension or extreme off-center load solutions in numerical solutions; compared to simplified methods equivalent to fixed weight allocation, this method, under the premise of satisfying strict force balance constraints, aims to achieve a higher tension... It can be flexibly adjusted according to actual working conditions, and has better engineering adaptability.

[0131] When a trolley is connected to only a single lug ( In the single-lug direct-connection configuration, the force balance equation of the suspension node automatically degenerates into the collinear constraint of the upper and lower ropes. The tension of the upper rope is equal to the corresponding tension of the lower rope. No additional branch processing logic is required, and the solution framework is completely consistent with that of the multi-lug configuration.

[0132] The coordinate relationships, connection relationships, and motion operations involved in this method will be explained below with reference to Figures 2 to 4.

[0133] As shown in Figure 2, this method uses a global rectangular coordinate system O-XYZ as a unified reference. The X-axis follows the direction of the gantry crane's main beam, i.e., the trolley's movement direction; the Y-axis follows the direction of the gantry crane's track, i.e., the overall movement direction of the gantry crane; and the Z-axis points vertically upwards. The center of gravity of the suspended load is denoted as G, and its global coordinates are... The position of the j-th car is denoted as... The position of the j-th hoisting node is denoted as The position of the i-th lug is denoted as The upper section of the rope connects to the trolley. With hoisting node Its length and tension are denoted as , respectively. and The lower section of the rope connects to the hoisting node. With lugs Its length and tension are denoted as , respectively. and The arrows in the diagram indicate the direction of movement of the trolley, the gantry crane, and the rope tension.

[0134] As shown in Figure 3(a), during the connection stage, the suspension bar is naturally suspended below the trolley, and the lower rope is connected to the lifting lug, but the system has not yet entered the load-bearing equilibrium state; at this time, the distance between the two ends of each lower rope is recorded and fixed as its physical length. No tension calculation is performed. As shown in Figure 3(b), during the straightening stage, the system solves the initial equilibrium configuration under the constraints of the lower rope length equality and the upper rope length inequality, so that the originally loose or not fully stressed rope gradually forms a load-bearing state, and the initial tension is further calculated.

[0135] As shown in Figure 4(a), when the trolley moves along the main beam, the trolley coordinates... The change alters the direction of the upper rope and the position of the hoisting node, which in turn transmits the changes to the lower rope and the attitude of the suspended load through the hoisting node. As shown in Figure 4(b), when the length of the upper rope changes... When changes occur, the hoisting node The feasible location changes accordingly, thus affecting the hanging lug. The method involves considering the center of gravity G of the suspended object and the tension distribution of each rope. Therefore, during the motion phase, this method requires iterative execution of geometric solutions, relaxation checks, and tension calculations as the trolley position and rope length control values ​​are updated.

[0136] The entire process of hoisting ship sections is divided into three stages: connection, straightening, and motion. The rope constraint states and mechanical solution tasks for these three stages are fundamentally different. In the connection stage, the rope is still in a relaxed connection state, and the system does not meet the static equilibrium condition. If tension is directly calculated at this time, it will generate an initial tension that does not exist. Therefore, this stage is only used to record and solidify the physical length of the lower rope section. In the straightening stage, the hoisted object gradually transitions from an unloaded state to an initial loaded state. It is necessary to solve for the initial equilibrium configuration with the lowest potential energy and the initial tension under the constraint of a fixed rope length. In the motion stage, the position of the trolley and the length of the upper rope section change continuously with the operation commands. The rope may switch between tension and relaxation states between different operation steps. Therefore, it is necessary to cyclically execute geometric solutions, relaxation detection, constraint reconstruction, and tension solutions.

[0137] By dividing the process into three stages, this scheme avoids problems such as excessive constraint application, spurious initial tension, and unstable motion state switching caused by mixing relaxation connection, initial tension, and motion update into the same equation set. This ensures that the constraint state changes throughout the lifting process are consistent with the actual physical process. Furthermore, this scheme integrates nonlinear geometric constraint optimization, Tikhonov regularized bounded least squares, and a unified global equation set into this three-stage framework. These address geometric nonlinearity, static indeterminacy, and the motion coupling of the two cranes, respectively, thereby systematically solving the problem of multiple overlapping mechanical difficulties in the collaborative lifting of two gantry cranes.

[0138] Relaxation detection and constraint reconstruction:

[0139] In actual hoisting operations, rope slack is a common physical phenomenon, especially when the trolley's lateral movement causes changes in the angle of some ropes. The occurrence of slack alters the system's constraint structure: slack ropes no longer provide geometric constraints and must be removed from the constraint equations to reconstruct the equation set using the active rope subset in order to obtain a physically consistent solution.

[0140] Therefore, the cascaded bidirectional relaxation propagation and constraint reconstruction process is as follows:

[0141] Upper rope slack criterion: If, after the geometric solution converges, the distance from a certain hoisting node to its trolley is significantly less than the set length of the upper rope (exceeding the 5 cm tolerance), then the upper rope is determined to be slack, the tension is set to zero, and the hoisting node loses the upper rope constraint.

[0142] Criterion for slack in the lower section of the rope: If the distance between the two ends of a lower section of the rope deviates from its physical length by more than 1 cm geometric tolerance, the rope is considered slack and the tension is set to zero.

[0143] The relaxed state needs to be cascaded in two directions to ensure the physical consistency of the system description.

[0144] Downward propagation: If the upper section of a hoisting rope becomes slack, the hoisting rope loses its upper section support, and all its lower section ropes become slack as a result, regardless of whether its geometric constraint error exceeds the tolerance.

[0145] Upward propagation: If all the lower ropes connected to a certain hoisting gantry are slack, then the hoisting gantry does not bear any downward load, and the tension of its upper ropes also drops to zero, and it is judged to be slack, regardless of whether its geometric constraint error exceeds the tolerance.

[0146] Both directions of propagation are indispensable: if downward propagation is ignored, the lower rope will still be incorrectly constrained when the upper rope is already slack, resulting in false constraints at the hoisting node position; if upward propagation is ignored, the upper rope will still be incorrectly tensioned when all the lower ropes are already slack, violating physical reality.

[0147] After determining the current set of active ropes, remove the constraint terms corresponding to all relaxed ropes from the geometric constraint equations and delete the corresponding columns from the coefficient matrix of the tension equations. Reconstruct and solve the geometric and mechanical equations with the active rope subset.

[0148] The relaxation detection and geometric solution are iterated alternately, with the convergence criterion being that the active rope set no longer changes. The upper limit of the iteration is 5 rounds. If the active set is not stable within this range, the result of the 5th round is taken and a convergence warning is output.

[0149] It should be noted that ropes deemed slack do not participate in the force balance equations and tension solutions in the current solution round, but are not removed from the system topology. At the beginning of each subsequent operation step, the system performs a candidate tension determination on slack ropes based on the updated trolley position, upper rope length, predicted load pose, and the state of the previous convergence step. If the predicted distance between the two ends of a slack rope reaches or approaches its physical length, and its associated hoisting rack has not been excluded by cascading propagation, the rope is added back to the candidate active set; subsequently, the rope length constraint is checked again through geometric solution to see if it can be satisfied simultaneously. If the check passes, the rope becomes an active rope and participates in the tension equations for this step; if the check fails, it remains slack with zero tension.

[0150] For a hoisting system where the upper rope and all its lower ropes are slack, first construct candidate positions for the hoisting system nodes based on the current trolley coordinates and the length of the upper rope, or use the position from the previous convergence step as the initial candidate value, and then determine whether the relevant lower ropes meet the conditions for restoring tension. This avoids the problem that slack ropes can never re-enter the active set because they are not involved in the current constraint solution.

[0151] It is worth noting that the moment a rope regains tension from a slack state is often accompanied by an impact load, the peak value of which may be much higher than the quasi-static calculation result. For such conditions, it is recommended to apply an appropriate impact amplification factor on top of the quasi-static tension given in the simulation, or to require operators to wind up the rope in steps at extremely low speeds in the operating procedures to suppress the generation of impact loads.

[0152] The cascaded bidirectional relaxation propagation mechanism takes into account that the upper rope, hoisting nodes, and lower rope form a continuous force chain, and the relaxation of any level of rope will affect the load transfer capacity of adjacent levels. If only unidirectional propagation is used, two types of non-physical states are likely to occur: first, the lower rope is completely relaxed while the upper rope is still retained as an active constraint, forming a false tension of "no load below, still under force above"; second, the upper rope is relaxed while the lower rope still participates in geometric constraints, forming a false constraint of "no support above, still constrained below". Therefore, this scheme sets up both downward and upward propagation. When the upper rope relaxes, its corresponding lower rope relaxes accordingly; when all the lower ropes of a hoisting system relax, its corresponding upper rope is also determined to be relaxed. Through bidirectional propagation and active rope set reconstruction, the relaxation rope can be prevented from continuing to participate in geometric constraints and tension solutions, improving the physical consistency and numerical stability of constraint switching during the motion phase.

[0153] Complete solution process for each step of the motion phase:

[0154] After each control variable update is completed, the system immediately performs a new round of equilibrium configuration and tension solution, which includes the following five steps.

[0155] Step 1: Using the equilibrium configuration from the previous step as the initial guess for hot start, solve for the new equilibrium configuration using nonlinear constraint optimization. In the first round, all rope constraints are used to ensure solution stability. From the second round onwards, ropes that have been identified as relaxed are removed, and constraints are constructed only with active ropes. This process is iterated alternately with relaxation detection until the set of active ropes converges.

[0156] Step 2: Based on the configuration determined after geometric iteration convergence, recalculate the unit direction vectors of all active ropes according to the set of active ropes, and construct the coefficient matrix of the tension linear equation system. and the right-hand vector We strictly eliminate columns corresponding to slack ropes to ensure that the dimensions of the equation system strictly correspond to the number of active unknowns.

[0157] Step 3: Apply Tikhonov regularization to the constructed system of equations and solve using bounded least squares. Under the premise of satisfying non-negativity constraints, obtain the tension of all active ropes. The tension of the relaxed ropes is strictly set to zero and is not included in the solution.

[0158] Step 4: Calculate the force balance residual Check whether it meets the convergence tolerance (force balance residual). Torque residual The system proceeds to step 5 after verifying that all solved tensions are non-negative. If the verification passes, the system continues to step 5. If the residual exceeds the limit or the active rope set is not stable within the limited number of rounds, the system does not terminate directly but instead initiates an adaptive re-solution mechanism: First, the current operation command is subdivided into smaller step sizes according to a preset ratio, and the previous safe equilibrium configuration is used as a hot start for re-solving; second, the active rope set is re-examined according to the relaxation criterion, and if necessary, the local iteration upper limit is increased or the initial guess of the hanging pile node is regenerated; the residual is calculated again and it is determined whether the tolerance is met.

[0159] Step 5: Establish the new equilibrium configuration and rope tensions as the current system state for use in the warm-up and safety assessment of the next operation step. Before issuing the next instruction, the operator can determine the safety margin based on the tension and attitude results of the current step and decide whether to adjust the subsequent operation plan.

[0160] When an operator plans multiple operations at once, each step is executed sequentially. Before proceeding to the next step, a complete geometric-mechanical solution is performed between each step to obtain the quasi-static equilibrium configuration. This step-by-step approach ensures that the result of each discrete operation step corresponds to the actual equilibrium state and avoids the risk of hot-start failure caused by large step size jumps.

[0161] This solution employs a step-by-step approach, prioritizing geometry before mechanics, and introduces Tikhonov regularization. The wire rope is treated as having inelastic geometric constraints, with its length remaining constant regardless of tension. Therefore, the orientation of the load and the location of the hoisting nodes can be independently determined by the rope length constraints. Once the geometric configuration is determined, the direction vectors of each rope are accordingly established, transforming the tension problem into a linear static equilibrium problem. This separation avoids ill-conditioned coupling caused by simultaneously solving for orientation and tension in the same nonlinear optimization, improving solution stability. For multi-lug and multi-crane systems, the static equilibrium equations typically have infinitely many non-negative feasible tension solutions. Tikhonov regularization here is not merely a general mathematical technique, but rather a load distribution criterion, guiding the solution from infinitely many feasible solutions to a solution that satisfies force balance and is closer to equilibrium bearing capacity, avoiding spurious negative tension, extreme off-center loading, or unnecessary local rope overload.

[0162] Safety assessment and results output:

[0163] After each solution step, output the rope utilization rate for all active ropes. ( For safety reasons, The system will trigger an overload alarm and output the overloaded rope number, overload level, and safety factor. (The safety factor for slack ropes is positive infinity, which is usually required by engineering specifications.) Rope utilization rate is defined as the ratio of actual tension to rated breaking force. A utilization rate not exceeding 1 is considered safe; exceeding 1 triggers an overload alarm and outputs the overloaded rope number, overload degree, and corresponding adjustment suggestions. Rope safety factor is defined as the ratio of rated breaking force to actual tension. The safety factor for slack ropes is positive infinity, and engineering specifications typically require the safety factor to be no less than the lower limit. The upper and lower rope sections are evaluated using the same method. In multi-lug configurations, the upper rope section bears the resultant force of all lower rope sections' tension, and its absolute value is usually higher than the tension of a single lower rope section; this should be a key consideration during design.

[0164] Comprehensive feasibility verification includes: attitude constraints of the suspended load (yaw angle) Pitch angle Roll angle All are within the allowable range of the process; if the predicted limit is exceeded, the operation plan should be adjusted in time. The following constraints apply: Lifting node height (the trajectory of the lifting node should always remain within the safe height range during movement to avoid interference and collision with ground obstacles, ship structures, or other equipment; a lower height alarm threshold can be set); Dual-machine load balance constraints (the simulation directly outputs the vertical resultant force borne by each crane; engineering specifications usually require that the load capacity of a single machine does not exceed the specified proportion of the rated lifting capacity, and the load ratio between the two machines should not be too disparate; when the predicted load ratio exceeds the allowable range, the lifting lug layout should be adjusted or the center of gravity coordinates should be re-estimated during the design phase, rather than relying on temporary operations for remedies); Operational executability constraints (the operation sequence planned in the simulation should be accurately executed by the actual equipment; the number of operation steps should not be too cumbersome; for schemes requiring high-precision multi-step coordination, the necessity of introducing an automated control system should be evaluated).

[0165] When an assessment shows that the utilization rate of a certain group of ropes is too high, the following approaches can be used to guide the improvement of the scheme: adjust the layout of the lifting lugs, moving the lifting lugs on the off-center side closer to the center of gravity or increasing the number of lifting lugs; change the rope length configuration, adjusting the relative proportion of the upper rope lengths to change the spatial position of the lifting nodes and thus adjust the tension distribution of each lower rope; optimize the trolley position, improving load balance by adjusting the lateral spacing of the trolleys. This closed loop based on quantitative analysis—"simulation solution - safety assessment - adjustment suggestions - re-solution"—runs through the entire lifecycle of the lifting scheme, directly providing quantitative theoretical basis for rope selection, lifting point layout optimization, and operational scheme review.

[0166] The purpose of this multi-dimensional safety assessment system is not only to output results such as utilization rate, safety factor, attitude angle, and dual-machine load ratio, but also to transform the assessment results into actionable adjustment suggestions for engineering projects. Specifically, when the rope utilization rate is too high, the system suggests adjusting the position of adjacent trolleys, the length of the upper rope section, or the layout of the lifting lugs; when the attitude angle of the suspended object exceeds the limit, the system suggests using differential trolley movement or differential rope retraction and release for attitude compensation; when the dual-machine load ratio is unbalanced, the system suggests reallocating lifting points, correcting the center of gravity input, or adjusting the synchronization strategy of the two cranes. Through a closed loop of "simulation solution - safety assessment - adjustment suggestions - re-solution," this solution can directly serve rope selection, lifting point layout optimization, operation plan review, and pre-operation contingency plan revision.

[0167] Implementation and verification under typical working conditions:

[0168] This solution will be specifically explained using the coordinated lifting of a certain hull section by two gantry cranes as an example. Figure 5 As shown. Mass of the suspended object. coefficient of inertia Equivalent total load The measured global coordinates of the centroid are The eccentricity in the Y direction is only -0.012m, and the segmented whole is approximately symmetrical about the Y axis.

[0169] Four lifting lugs are arranged on the suspended object, divided into two groups in the X direction: the two lugs in the front group (lug 0 and lug 1) are located at... At this location, the span in the Y direction is The posterior group of two ears (ear 2 and ear 3) are located in At this location, the span in the Y direction is The height of all four ears is uniform. The center of gravity is roughly level with the height of the center of gravity (6.527 m). The asymmetrical layout of the front group with a large span and the rear group with a small span directly determines the difference in load distribution between the two gantry cranes.

[0170] Corresponding to the layout of the lifting lugs, a dual-gantry crane collaborative scheme is adopted: Gantry crane A (track located at...) Two trolleys are configured, one facing each of the two front lugs, with each trolley having an upper rope length of 12.00m; Gantry crane B (track located at...) Similarly, two trolleys are configured, directly opposite the two lugs of the rear group, with the upper rope length also being 12.00m. Each trolley is directly connected to a lower rope via a hoisting node to the corresponding lug (direct connection configuration). The entire system consists of 4 upper ropes and 4 lower ropes, with the trolley main beam height... .

[0171] After the connection phase is completed, the length snapshots of each lower rope segment are fixed. This process remains unchanged thereafter. No mechanical solutions are performed during the connection phase.

[0172] After entering the straightening stage, the solver aims to minimize the Z-coordinate of the center of gravity and determines the spatial position of each hoisting node and the orientation of the suspended load using a sequential quadratic programming method. An initial guess is generated based on the simplified assumption that "all ropes are vertical," and the initial estimate of the center of gravity height is... The initial height of each hoisting node is taken as (Slightly below the tautness threshold, ensuring the start point is inside the constraint). After iterative convergence, the height of all four hanging nodes is positioned at... The geometric relationship between the height of the trolley (25.00m) and the length of the upper rope (12.000m) is completely consistent, indicating that the solution in the straightening stage is normal.

[0173] Due to the adoption of a direct connection configuration ( According to the force balance equation of the hoisting nodes, the upper rope segment and the corresponding lower rope segment are collinear, and their tension values ​​are completely equal. The tension solution results are as follows: On side A of the gantry crane, the rope tension corresponding to trolley A-0 is 15.71tf, and the rope tension corresponding to trolley A-1 is 15.83tf; on side B of the gantry crane, the rope tension corresponding to trolley B-0 is 11.71tf, and the rope tension corresponding to trolley B-1 is 11.77tf. The simulated total load is accurately restored to 55.0tf, and the total load conservation accuracy reaches within 0.62%. The tension on side A (approximately 15.7~15.8tf) is significantly higher than that on side B (approximately 11.7tf), with a difference of approximately 4tf, which is highly consistent with the physical logic that the center of gravity of the hoisted object is slightly biased towards the front group of lifting lugs (side A). Comparing the simulation results with the analytical solution of uniform distribution of the four vertical ropes, the maximum deviation of the tension of each rope is 2.09%, which meets the engineering accuracy requirements. The magnitudes of the X and Y component forces of each rope are only about 1 × 10⁻ 4 The force tf is about five orders of magnitude smaller than the Z-direction component, which is completely consistent with the physical expectation of "vertical", indirectly verifying the convergence quality of the geometric solution.

[0174] Entering the motion phase, simulating the vertical ascent of the suspended object: the four upper ropes are simultaneously shortened, with each step involving rope retraction. Time step The simulation consisted of 5 steps, with a total rope take-up of 1.000m. After each step, the system automatically retried the complete solution process, using the equilibrium configuration of the previous step as the initial guess for hot start. Simulation results show that the center of gravity of the suspended object remained constant throughout the X and Y directions, while the Z direction rose precisely by 0.200m per step, accumulating to a total rise of 1.000m over five steps, strictly corresponding to the rope take-up step length. This achieved pure Z-direction translation without horizontal drift or attitude deflection. The change in tension of each rope throughout the entire ascent was within 1×10⁻⁻⁻⁶. 6 The magnitude is on the order of tf, which can be considered zero. The physical explanation for this result is that synchronous rope winding causes the suspended object to translate along the Z-axis; the rope direction vector does not change with height, therefore the coefficient matrix of the force balance equation is... With the right-hand vector Both remain unchanged, and the tension solution naturally remains constant, perfectly matching the predictions of the theoretical model. In the multi-step iterations, the deviation of the suspended object's trajectory and the change in tension are both within 1×10⁻⁻⁻⁶. 6 The magnitude indicates that the proposed solution has good numerical robustness and is suitable for repeated use in actual engineering projects to support incremental planning.

[0175] A rigid-flexible coupling simulation system for hoisting ship hull sections includes:

[0176] The coordinate system and pose modeling unit is configured to: establish a global rectangular coordinate system, use Euler angles to describe the spatial attitude of the suspended object, and use the body coordinate system to parameterize the global coordinates of each lug;

[0177] The three-stage process management unit is configured to divide the entire hoisting process into three stages: connection stage, straightening stage, and motion stage. Each stage has different constraint structures and solution tasks.

[0178] The solution unit is configured to integrate the force balance equations of the suspended object and the force balance equations of each suspension node into a unified linear equation system with respect to the tension of all active ropes in each solution step. The Tikhonov regularized bounded least squares method is used to solve the equations under non-negative constraints, so that the solution results tend to uniform load distribution under the premise of satisfying force balance.

[0179] The relaxation detection and constraint reconstruction unit is configured to: identify relaxed ropes based on geometric criteria, update the active rope set through cascaded bidirectional propagation between the upper rope, the hoisting node and the lower rope, reconstruct the equation set after removing the geometric constraints and tension equations corresponding to the relaxed ropes, and iterate alternately through relaxation detection and geometric solution until the active rope set converges.

[0180] The evaluation and verification unit is configured to output safety evaluation indicators for the active rope after each solution step and to comprehensively verify the attitude of the suspended object and the load balance of each machine.

[0181] The entire hoisting process is divided into three stages: connection, straightening, and movement. This avoids the problem of excessive constraint application caused by mixing slack and taut ropes in the same set of equations, ensuring that the rope state switching is consistent with the actual physical process. Through the solution, a unique solution that satisfies force balance and tends towards uniform load distribution is automatically selected from infinitely many feasible tension solutions, fundamentally avoiding the risk of severe overload of individual ropes due to unreasonable tension distribution. At the same time, through a cascaded bidirectional relaxation propagation and constraint reconstruction mechanism based on geometric criteria, the influence of slack ropes on constraint equations and mechanical equations is automatically identified and eliminated, further ensuring that the input conditions for static indeterminate solutions are strictly consistent with the actual physical state of the system. The combined application of the above technical means enables this scheme to automatically obtain physically self-consistent and load-balanced tension solutions without human intervention, providing a reliable quantitative basis for the safety assessment of multi-machine hoisting projects.

[0182] Those skilled in the art will understand that embodiments of this solution can be embodied as a method, system, or computer program product. Therefore, this solution can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this solution can also take the form of a computer program product implemented on one or more computer-readable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0183] This solution is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this solution. It should be understood that each block in the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to create a machine such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, can implement one or more blocks in the flowchart and / or one or more blocks in the block diagram, specifying the function.

[0184] These computer program instructions may also be stored in a computer-readable storage medium capable of directing a computer or other programmable data processing device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium generate an article of manufacture containing instruction means. These instruction means are used to implement the functions specified in one or more flowcharts and / or one or more blocks of a block diagram.

[0185] Furthermore, these computer program instructions can be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to generate a computer-implemented processing procedure. Thus, the instructions that execute on the computer or other programmable equipment will provide steps for implementing one or more processes of the flowchart and / or one or more blocks of the block diagram that specify the functionality.

[0186] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A rigid-flexible coupling simulation method for hoisting ship hull sections, characterized in that, Includes the following steps: Step S1: Establish a global rectangular coordinate system, use Euler angles to describe the spatial attitude of the suspended object, use the body coordinate system to parameterize the global coordinates of each lifting lug, and abstract the hoisting system into a multi-level force transmission structure of "trolley-upper rope-lifting block node-lower rope-lifting lug-suspended object"; Step S2: Divide the entire hoisting process into three stages: connection stage, straightening stage, and motion stage. Each stage has different constraint structures and solution tasks. During the connection phase, the spatial distance between the two ends of each lower rope segment is recorded as its physical length and fixed while the rope is in a slack state, without performing mechanical solutions. During the straightening stage, with the goal of minimizing the height of the center of gravity of the suspended object, the initial equilibrium configuration and initial rope tension of the system are solved by nonlinear constraint optimization under the constraints of the lower rope length equality and the upper rope length inequality. During the motion phase, as the control variables are gradually updated, geometric solutions, relaxation detection, and tension solutions are cyclically executed. Step S3: In each solution step, the force balance equation of the suspended object and the force balance equation of each suspension node are integrated into a unified linear equation system about the tension of all active ropes, and solved under non-negative constraints, so that the solution results tend to uniform load distribution under the premise of satisfying force balance. Step S4: Identify slack ropes based on geometric criteria, update the active rope set through cascading bidirectional propagation between the upper rope, the hoisting node and the lower rope, reconstruct the equation set after removing the geometric constraints and tension equation terms corresponding to the slack ropes, and iterate alternately through slack detection and geometric solution until the active rope set converges. Step S5: After each solution step is completed, output safety assessment indicators for the active rope and comprehensively verify the attitude of the suspended object and the load balance of each machine.

2. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, The global rectangular coordinate system takes the ground reference point of the main assembly platform as the origin, with the X-axis along the direction of the gantry crane main beam, the Y-axis along the direction of the gantry crane track, and the Z-axis vertically upward. The body coordinate system takes the center of gravity of the suspended object as its origin, and the coordinates of each lifting lug in the body coordinate system are calculated and fixed when the connection stage is completed. The global coordinates of the i-th lug at any given time are calculated by the following formula: ； in For the global coordinates of the center of gravity of the suspended object, Let ZYX be the order Euler angle rotation matrix. Let be the fixed coordinates of the i-th lug in the body coordinate system.

3. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, The rope length fixing process during the connection phase is as follows: the connection operation is performed when the rope is fully slack and the hoist is vertically suspended directly below the trolley. The initial coordinates of the hoist nodes are determined by offsetting the trolley coordinates downwards by the length of the upper rope segment. The physical length of each lower rope segment is fixed according to the spatial distance between the two endpoints, as shown in the following formula: ； in, Let be the fixed physical length of the i-th lower segment of the rope; For the connection phase The initial global coordinates of the first hoist lug; Let j(i) be the initial coordinates of the j(i)th hoisting node connected to the i-th lower rope segment; j(i) represents the hoisting number to which the i-th lower rope segment belongs. This represents the total number of ropes in the next section.

4. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, With the objective of minimizing the Z-coordinate of the suspended load's center of gravity, a nonlinear constrained optimization problem is constructed under the constraints of equality of the lower rope length and inequality of the upper rope length. This problem is solved iteratively using a sequential quadratic programming method. The initial conjecture is generated based on the simplified assumption that "all ropes are vertical," and the estimated height of the suspended load's center of gravity is: ； in, This represents the initial estimate of the height of the center of gravity of the suspended load. Indicates the number of active trolleys involved in the hoisting operation; This represents the height of the main beam of the j-th car; This represents the initial length of the j-th upper rope segment set during the connection phase; Indicates the total number of rope segments in the lower section; This represents the physical length of the i-th lower rope segment after it has solidified during the connection stage; After the geometric solution converges, the unit direction vectors of each rope are substituted into the tension equations, and the initial tension is solved using the Tikhonov regularized bounded least squares method.

5. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, The solution strategy for the tension equations is as follows: equilibrium of the suspended load and the structure of the active lifting platform nodes. A system of linear equations; the tension is ensured to be non-negative by using bounded least squares solution. When the system is statically indeterminate, Tikhonov regularization is introduced, with the regularization coefficient and the expected uniform tension of each rope as the basis. The solution is guided towards uniform load distribution; when each hoisting bar is connected to only a single lifting lug, the hoisting bar node equations degenerate into three-point collinear constraints.

6. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, The cascaded bidirectional relaxation propagation and constraint reconstruction process is as follows: After the geometric solution converges, if the distance from the hoisting node to the trolley is less than the set length of the upper rope and exceeds the preset tolerance, the upper rope is determined to be slack and the tension is set to zero. If the distance between the two ends of a lower section of rope deviates from its physical length by more than the preset tolerance, the rope is determined to be slack and the tension is set to zero. When the upper section of the rope is slack, all the lower sections of the rope connected to the hoisting machine will also be slack; when all the lower sections of the rope of a hoisting machine are slack, the upper section of the rope of the hoisting machine will also be considered slack. By removing the geometric constraints and tension equations corresponding to the slack rope, the equations are reconstructed and solved again based on the remaining active rope.

7. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 6, characterized in that, The geometric tolerance for relaxation detection is as follows: the relaxation tolerance of the upper rope segment and the relaxation tolerance of the lower rope segment are both set values; when the number of iterations of relaxation detection and geometric solution exceeds the upper limit, a convergence warning is output, and the convergence state of the active rope set is used as one of the judgment conditions of the adaptive re-solution mechanism.

8. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 1, characterized in that, The single-step solution during the motion phase includes the following steps: Step S201: Using the equilibrium configuration from the previous step as the initial guess for hot start, perform geometric solution and relaxation detection iterations until the active rope set converges; Step S202: Based on the converged set of active ropes, recalculate the unit direction vectors of all active ropes and construct the coefficient matrix A and the right-hand vector b of the tension linear equation system. Step S203: Apply Tikhonov regularization bounded least squares solution to obtain all active rope tensions, and relax the rope tensions to strictly zero. Step S204: Calculate the force balance residuals and verify the non-negativity constraints; Step S205 establishes the new equilibrium configuration and rope tension as the current system state for use in the next operation step of hot start-up and safety assessment.

9. The rigid-flexible coupling simulation method for hoisting ship sections as described in claim 8, characterized in that, When the force balance residual, torque residual, or active rope set convergence state does not meet the preset conditions, an adaptive re-solution mechanism is activated, including reducing the step size, using the successful convergence step as a hot start for re-solution, increasing the local iteration upper limit, or regenerating the initial guess of the hoisting node. If multiple re-solutions still do not meet the conditions, an infeasibility alarm is output.

10. A rigid-flexible coupling simulation system for hoisting ship sections, used to implement the method described in any one of claims 1-9, characterized in that, include: The coordinate system and pose modeling unit is configured to: establish a global rectangular coordinate system, use Euler angles to describe the spatial attitude of the suspended object, and use the body coordinate system to parameterize the global coordinates of each lug; The three-stage process management unit is configured to divide the entire hoisting process into three stages: connection stage, straightening stage, and motion stage. Each stage has different constraint structures and solution tasks. The solution unit is configured to integrate the force balance equations of the suspended object and the force balance equations of each suspension node into a unified linear equation system about the tension of all active ropes in each solution step, and solve it under non-negative constraints so that the solution results tend to uniform load distribution under the premise of satisfying force balance. The relaxation detection and constraint reconstruction unit is configured to: identify relaxed ropes based on geometric criteria, update the active rope set through cascaded bidirectional propagation between the upper rope, the hoisting node and the lower rope, reconstruct the equation set after removing the geometric constraints and tension equations corresponding to the relaxed ropes, and iterate alternately through relaxation detection and geometric solution until the active rope set converges. The evaluation and verification unit is configured to output safety evaluation indicators for the active rope after each solution step and to comprehensively verify the attitude of the suspended object and the load balance of each machine.

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