Quantitative analysis method and system for superconducting sub-cable stick-slip bending behavior under electromagnetic force load

Abstract

The application discloses a kind of quantitative analysis method and system of superconducting sub-cable viscous-slippage bending behavior under electromagnetic force load, belong to fusion superconducting magnet and in-tube cable conductor mechanics analysis field.This method establishes three-dimensional slender rod balance equation containing electromagnetic force term and coulomb friction term under Frenet local coordinate system, introduces pre-stretching boundary condition as initial value, solves full arc length internal force and contact force distribution in combination with four-order Runge-Kutta numerical propulsion strategy, determines viscous-slippage bending state according to viscous stress and slippage stress relationship and calculates sub-cable total bending moment, determines turning point to obtain unloading curve by viscous-slippage determination function in unloading stage, and then obtains dissipation energy by the area surrounded by loading and unloading curve.The application realizes multi-scale quantitative analysis of electromagnetic-friction coupling, can accurately predict the evolution of superconducting sub-cable macroscopic bending moment, equivalent bending stiffness and cyclic hysteresis energy dissipation characteristics under electromagnetic force and tensile-bending combined load.

Description

Technical Field

[0001] This invention belongs to the field of mechanical analysis technology of fusion superconducting magnets and in-tube cable conductors, and particularly relates to a quantitative analysis method and system for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force load. Background Technology

[0002] Multi-level superconducting helical sub-cables combine high load-bearing capacity with flexible deformation capabilities. The electromagnetic force compression effect and the viscous-slip coupling behavior between strands generated during their service are the core mechanical foundation for the safe and stable operation of fusion superconducting magnets and in-tube cable conductors. This type of structure relies on multi-level helical stranding to form a complex multi-scale configuration. The mechanical responses of strands at different levels are nested and coupled with each other. At the same time, microscopic behaviors such as the evolution of interfacial contact pressure, Coulomb friction slip, and cyclic friction energy dissipation are highly correlated with the macroscopic bending stiffness and hysteretic mechanical properties of the sub-cable, which brings great challenges to the accurate prediction of its service mechanical behavior and the optimization and control of its structural performance.

[0003] Three-dimensional thin-bar theory is a classic theoretical tool for analyzing the internal forces, contact, and deformation behavior of helical cables. It can fully characterize the three-dimensional internal force and torque balance relationship of a single strand and is applicable to the micromechanical analysis of superconducting multi-stage helical sub-cables. However, existing theoretical models do not couple electromagnetic force distribution loads, making it impossible to quantify the contribution of Lorentz forces to changing interfacial contact pressure and regulating the critical curvature of the viscosity-slip transition. This deficiency makes it difficult for the models to accurately reflect the real mechanical response of superconducting sub-cables under strong electromagnetic service environments. At the same time, existing models cannot quickly quantify and evaluate the mechanism by which the geometric parameters of superconducting sub-cables under actual service conditions affect stick-slip bending and frictional losses. Furthermore, existing experimental testing methods are insufficient to precisely capture the multi-scale stick-slip evolution and interfacial contact stress distribution characteristics of superconducting sub-cables under electromagnetic load coupling; relying solely on experimental methods cannot meet the needs of engineering analysis. At the theoretical modeling level, the contact state between strands in a multi-stage helical structure dynamically changes with bending deformation. The interface delineation between the viscous and slip zones depends on the nonlinear coupling relationship between contact pressure, friction coefficient, and bending curvature. This presents a dual challenge in theoretical derivation and numerical implementation when simultaneously introducing electromagnetic distributed loads and Coulomb friction terms within the existing framework of slender rod theory to achieve stable numerical solutions, thus hindering the establishment of multi-field coupled analysis methods. Therefore, it is urgent to systematically reveal the control mechanism of electromagnetic loads, geometric configurations, and interfacial friction parameters on the bending mechanical properties of multi-stage superconducting sub-cables through slender rod theoretical modeling and numerical analytical methods. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a quantitative analysis method and system for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force load, thereby resolving the issues existing in the prior art.

[0005] In a first aspect, to achieve the above objectives, the present invention provides a quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading, comprising the following steps: S1. Establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. S2. Based on the aforementioned thin rod equilibrium equation, and combined with the fourth-order Runge-Kutta numerical propulsion strategy, solve the distribution of internal forces and contact forces of the strand within the entire arc length. Determine the viscous-slip bending state at each location based on the relationship between viscous stress and slip stress, and calculate the correspondence between the total bending moment of the sub-cable and the current bending curvature based on the determination results. S3. During the unloading phase, based on the determination function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, the inflection point of the viscous-slip state during the unloading process is determined, the bending moment-curvature curve of the unloading section is obtained, and then the energy dissipation per unit length of sub-cable during the loading and unloading cycle is calculated.

[0006] Optionally, the process of establishing the equilibrium equations for the thin rod includes: in the Frenet local coordinate system, using the arc coordinates of the strand as variables, establishing the equilibrium relationship between the normal curvature, subnormal curvature, and tangential torsion of the deformed strand and the internal forces and internal moments distributed along the arc length of the strand, and introducing an electromagnetic force load term and a Coulomb friction coefficient term into this equilibrium relationship. The electromagnetic force load term is expressed as the product of current and magnetic field, and the Coulomb friction coefficient term is used to express the proportional relationship between the contact force of the normal and subnormal lines and the tangential friction force.

[0007] Optionally, the process of introducing boundary conditions under the initial pre-stretch state includes: setting the initial tangential strain of the strand and the initial normal contact force caused by tension, substituting the cross-sectional area of ​​the strand as a parameter, solving the internal force and internal moment components at each arc length position under the uniaxial tension state along the arc length of the strand, and using the internal force and contact force values ​​at the neutral axis position as the initial values ​​for solving the bending deformation.

[0008] Optionally, the determination process of viscous-slip bending state includes: obtaining slip stress based on tangential strain caused by friction; using the relationship between the distance from the center line of the strand to the neutral axis of the sub-cable under the current bending curvature and viscous stress and slip stress as the determination basis; using the critical bending curvature as the dividing threshold; when the stress on the strand is less than or equal to the stress value corresponding to the critical bending curvature, it is determined to be in viscous state; otherwise, it is determined to be in slip state; and calculating the bending stiffness of the strand under the two states respectively.

[0009] Optionally, the calculation process for the correspondence between the total bending moment of the sub-cable and the current bending curvature includes: dividing the winding angle into several discrete segments within a set angle range, calculating the bending stiffness of each discrete segment in the viscous state and the bending stiffness in the slip state, projecting and summing the bending stiffness of each discrete segment according to the symmetry of the first-order helix, and then multiplying it by the current bending curvature to obtain an explicit expression of the total bending moment of the sub-cable.

[0010] Optionally, the process of constructing the determination function includes: assuming that the sub-cable is in a completely viscous state during the unloading process, calculating the sub-cable bending moment corresponding to the cumulative unloading curvature, comparing the bending moment with the peak coordinate of the bending moment-curvature curve, determining that the state is viscous when the unloading bending moment in the assumed completely viscous state is less than the peak bending moment, and determining that the state has entered a stick-slip state when the unloading bending moment in the assumed completely viscous state is greater than or equal to the peak bending moment, and using the bending curvature corresponding to the inflection point of entering the stick-slip state as the critical curvature of the unloading stage; the process of calculating the dissipated energy includes: integrating and subtracting the loading curve and the unloading curve within the bending curvature interval to obtain the dissipated energy per unit length of sub-cable in a complete loading and unloading cycle.

[0011] Secondly, the present invention also provides a quantitative analysis system for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading, used to implement a quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading, the system comprising: The initial condition setting module is used to establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. The stick-slip bending calculation module is loaded to solve the distribution of internal forces and contact forces of the strand over the entire arc length based on the balance equation of the thin rod and combined with the fourth-order Runge-Kutta numerical propulsion strategy. The stick-slip bending state at each position is determined according to the relationship between viscous stress and slip stress, and the correspondence between the total bending moment of the sub-cable and the current bending curvature is calculated based on the determination result. The unloading energy consumption calculation module is used to determine the inflection point of the viscous-slip state during the unloading phase based on the judgment function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, obtain the bending moment-curvature curve of the unloading section, and then calculate the energy dissipation per unit length of sub-cable during the loading and unloading cycle.

[0012] Thirdly, the present invention also provides a computer terminal device, comprising: One or more processors; A memory, coupled to the processor, for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the steps of the quantitative analysis method for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force load in the first aspect above.

[0013] Fourthly, the present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the steps of the quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force load as described in the first aspect above.

[0014] Fifthly, the present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force load as described in the first aspect above.

[0015] Compared with the prior art, the present invention has the following advantages and technical effects: This invention provides a quantitative analysis method and system for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force loading. A three-dimensional numerical calculation model of a thin rod integrating electromagnetic distributed load and Coulomb friction is established, fully considering the multi-stage helical stranding configuration and strand contact arrangement characteristics of the superconducting sub-cable. This achieves integrated and simultaneous analysis of macroscopic bending hysteresis behavior and microscopic slip contact field, enabling the solution of the complete mechanical evolution law of the sub-cable under electromagnetic-tension-bending composite loads. This invention clarifies the multi-parameter control mechanism of electromagnetic load, helical lay-up angle, pre-tension strain, and Coulomb friction coefficient on the overall bending stiffness, viscosity-slip transition critical curvature, and local slip stress and contact equivalent stress distribution of the sub-cable. It quantitatively reveals the synergistic evolution law of macroscopic hysteretic deformation and interfacial friction slip behavior under the coupled action of electromagnetic force and mechanical load, and elucidates the key influences of interfacial contact compression and frictional dissipation on the cyclic bending service performance of the superconducting sub-cable. This invention improves computational efficiency while ensuring computational accuracy and fully preserving the microscopic information of strand slip stress, contact stress, and cyclic friction energy consumption. It establishes a complete evaluation system for the mechanical properties of superconducting sub-cables, covering macroscopic bending stiffness, critical slip curvature, cyclic friction energy consumption, and local contact stress. It also improves the theoretical system for multi-field coupling analysis of stick-slip bending of multi-stage superconducting helical sub-cables under strong electromagnetic service conditions. Attached Figure Description

[0016] 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 undue limitation of the invention. In the drawings: Figure 1This is a schematic diagram of the viscosity-slip bending behavior of the sub-cable under external load in an embodiment of the present invention. (a) shows the complex external loads borne by the helical sub-cable, including thermal mismatch strain, electromagnetic force and frictional contact force; (b) shows the multi-level frictional contact transmission behavior inside the sub-cable, where the point contact behavior exhibits a discrete periodic distribution, the line contact behavior exhibits a continuous distribution, and the inter-line frictional interaction exhibits a step-by-step transmission characteristic. Figure 2 This is a schematic diagram of the force analysis of a single strand of wire according to an embodiment of the present invention; Figure 3 This is a schematic diagram illustrating the solution process for the nonlinear bending response and energy dissipation of the sub-cable in an embodiment of the present invention. Detailed Implementation

[0017] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0018] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0019] Example 1 like Figure 3 As shown, this embodiment provides a quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force load, including: S1. Establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. S2. Based on the aforementioned thin rod equilibrium equation, and combined with the fourth-order Runge-Kutta numerical propulsion strategy, solve the distribution of internal forces and contact forces of the strand within the entire arc length. Determine the viscous-slip bending state at each location based on the relationship between viscous stress and slip stress, and calculate the correspondence between the total bending moment of the sub-cable and the current bending curvature based on the determination results. S3. During the unloading phase, based on the determination function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, the inflection point of the viscous-slip state during the unloading process is determined, the bending moment-curvature curve of the unloading section is obtained, and then the energy dissipation per unit length of sub-cable during the loading and unloading cycle is calculated.

[0020] Furthermore, the process of establishing the equilibrium equations for the thin rod includes: in the Frenet local coordinate system, using the arc coordinates of the strands as variables, establishing the equilibrium relationship between the normal curvature, subnormal curvature, and tangential torsion of the deformed strands and the internal forces and internal moments distributed along the arc length of the strands; and introducing an electromagnetic force load term and a Coulomb friction coefficient term into this equilibrium relationship. The electromagnetic force load term is expressed as the product of current and magnetic field, and the Coulomb friction coefficient term is used to express the proportional relationship between the contact force of the normal and subnormal lines and the tangential friction force.

[0021] Specifically, the implementation process of this embodiment includes: This numerical method mainly consists of two parts: viscous-slip bending response and energy dissipation calculation caused by loading and unloading. Numerical modeling is performed on each part, and nonlinear bending behavior is analyzed by numerically solving the theoretical equilibrium equation of the thin rod.

[0022] Viscous-slip bending response: The governing equations for a primary sub-cable under initial pre-tension and electromagnetic force load are given. Figure 2 As shown, in the Frenet coordinate system The equilibrium equation for any strand in the sub-cable is expressed as: ; Where s is the arc coordinate of the line, These represent the normal, binormal curvature, and tangential torsion of the deformed strand. The overline indicates the parameters after deformation. The internal force along the line of the stock. For internal torque, These are the normal and subnormal contact forces and the tangential frictional force. For the laying angle of the stock line, Rw is the twist angle of the strand, and Rw is the radius of the strand. IB represents the Coulomb friction coefficient, and IB represents the product of the current and the magnetic field, which is the electromagnetic force load.

[0023] Furthermore, the process of introducing boundary conditions under the initial pre-tension state includes: setting the initial tangential strain of the strand and the initial normal contact force caused by tension, substituting the cross-sectional area of ​​the strand as a parameter, solving the internal force and internal moment components at each arc length position under the uniaxial tension state along the arc length of the strand, and using the internal force and contact force values ​​at the neutral axis position as the initial values ​​for solving the bending deformation.

[0024] Specifically, the implementation process of this embodiment includes: The boundary conditions corresponding to pre-stretching are expressed as follows: ; in, and The initial normal contact force is caused by the initial tangential strain and tension of the strand. The cross-sectional area of ​​the strand is given by the subscript. This indicates the magnitude of the internal forces and contact forces at the neutral axis. First, solve for the corresponding internal force and moment components under uniaxial tension. , , as well as Substituting the above boundary conditions, we can obtain This is used as the initial condition for solving the bending deformation problem, and then combined with the fourth-order Runge-Kutta method to complete the progressive solution. Furthermore, we define... The tangential strain caused by friction can then be used to obtain the slip stress: .

[0025] Furthermore, the determination process for the viscous-slip bending state includes: obtaining the slip stress based on the tangential strain caused by friction; using the relationship between the distance from the center line of the strand to the neutral axis of the sub-cable under the current bending curvature and the viscous stress and slip stress as the determination basis; using the critical bending curvature as the dividing threshold; when the stress on the strand is less than or equal to the stress value corresponding to the critical bending curvature, it is determined to be in a viscous state; otherwise, it is determined to be in a slip state; and the bending stiffness of the strand is calculated under both states.

[0026] Specifically, the implementation process of this embodiment includes: Define critical curvature To determine whether the strand at this point is in a viscous or slipping bending state, consider the viscous stress. and slip stress The expression is given as follows: ; in, This represents the distance from a point on the center line of the main cable to the neutral axis of the sub-cable.

[0027] Furthermore, the calculation process for the correspondence between the total bending moment of the sub-cable and the current bending curvature includes: dividing the winding angle into several discrete segments within a set angle range, calculating the bending stiffness of each discrete segment in the viscous state and the bending stiffness in the slip state, projecting and summing the bending stiffness of each discrete segment according to the symmetry of the first-order helix, and then multiplying it by the current bending curvature to obtain an explicit expression of the total bending moment of the sub-cable.

[0028] Specifically, the implementation process of this embodiment includes: The expression for the total bending moment of the sub-cable can be derived using the projection method: ; in, This represents the current curvature. This refers to the number of spirals inside the primary sub-cable. This represents the bending stiffness corresponding to the current bending curvature. Based on the symmetry of the first-order helix, the winding angle... The range is: number of discrete segments n The value is set to 2000 to ensure calculation accuracy. The superscripts "stick" and "slip" represent the strands in viscous and slip states, respectively. The bending stiffness at the point. The subscript w indicates a single strand.

[0029] By stress and Integrating the cross-section of the strand yields the bending stiffness under two bending conditions. and -slip explicit expression: .

[0030] Furthermore, the process of constructing the determination function includes: assuming that the sub-cable is in a completely viscous state during unloading, calculating the sub-cable bending moment corresponding to the cumulative unloading curvature, comparing this bending moment with the peak coordinate of the bending moment-curvature curve, determining that the state is viscous when the unloading bending moment in the assumed completely viscous state is less than the peak bending moment, and determining that the state has entered a stick-slip state when the unloading bending moment in the assumed completely viscous state is greater than or equal to the peak bending moment, and using the bending curvature corresponding to the inflection point of entering the stick-slip state as the critical curvature of the unloading stage; the process of calculating the dissipated energy includes: integrating and subtracting the loading curve and the unloading curve within the bending curvature interval to obtain the dissipated energy per unit length of sub-cable in a complete loading and unloading cycle.

[0031] Specifically, the implementation process of this embodiment includes: Besides the loading phase, the sub-cable often undergoes loading and unloading processes during actual service. The energy dissipation of the multi-stage helical structure during unloading bending is crucial for a deeper understanding of friction-induced energy dissipation. During the unloading phase, because the sub-cable's viscosity-slip bending response exhibits path dependence, it is necessary to define a viscosity-slip criterion function. To characterize the current bending state: ; in, Assuming a fully viscous state, the cumulative unloading curvature The corresponding sub-cable bending moment, satisfy ,in It is defined as the peak coordinate of the moment-curvature curve. The critical bending curvature is defined as the point at which the transition from the fully viscous phase to the stick-slip phase: ; The moment-curvature curve of the unloading section can be obtained using the above determination function. Based on the above derivation, the evolution law of the nonlinear bending stiffness of the sub-cable during viscous-slip bending with respect to external load can be obtained. Furthermore, the area of ​​the closed region enclosed by the loading and unloading paths of the bending curvature-moment is the energy dissipated per unit length of the first-stage sub-cable in a complete loading and unloading cycle. : ; in, and. The dots represent the loading and unloading curves, respectively. Based on the three formulas for the current bending state, critical bending curvature, and dissipated energy, the energy dissipation of the primary sub-cable due to friction during the loading and unloading cycle is calculated.

[0032] Figure 1 A schematic diagram for the viscous-slip bending behavior of the sub-cable under external load is shown. (a) shows the complex external loads borne by the helical sub-cable, including thermal mismatch strain, electromagnetic force and frictional contact force; (b) shows the multi-level frictional contact transmission behavior inside the sub-cable, where the point contact behavior exhibits a discrete periodic distribution, the line contact behavior exhibits a continuous distribution, and the inter-line frictional interaction exhibits a step-by-step transmission characteristic.

[0033] Example 2 Based on the same general inventive concept, this invention also provides a quantitative analysis system for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading. The quantitative analysis system for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading provided by this invention is described below. The quantitative analysis system for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading described below can be referred to in correspondence with the quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading described above. The system includes: The initial condition setting module is used to establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. The stick-slip bending calculation module is loaded to solve the distribution of internal forces and contact forces of the strand over the entire arc length based on the balance equation of the thin rod and combined with the fourth-order Runge-Kutta numerical propulsion strategy. The stick-slip bending state at each position is determined according to the relationship between viscous stress and slip stress, and the correspondence between the total bending moment of the sub-cable and the current bending curvature is calculated based on the determination result. The unloading energy consumption calculation module is used to determine the inflection point of the viscous-slip state during the unloading phase based on the judgment function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, obtain the bending moment-curvature curve of the unloading section, and then calculate the energy dissipation per unit length of sub-cable during the loading and unloading cycle.

[0034] Example 3 (equivalent to Example 2 above) In this embodiment, a computer terminal device is provided, including: One or more processors; A memory, coupled to the processor, for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the steps of the quantitative analysis method for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force loads.

[0035] In this embodiment, a computer-readable storage medium is also provided, on which a computer program is stored. When the computer program is executed by a processor, it implements the steps of the quantitative analysis method for the viscous-slip bending behavior of the superconducting sub-cable under electromagnetic force load described above.

[0036] In this embodiment, a computer program product is also provided, including a computer program that, when executed by a processor, implements the steps of the quantitative analysis method for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force load described above.

[0037] This invention provides a quantitative analysis method and system for the viscous-slip bending behavior of superconducting sub-cables under electromagnetic force loading. A three-dimensional numerical calculation model of a thin rod integrating electromagnetic distributed load and Coulomb friction is established, fully considering the multi-stage helical stranding configuration and strand contact arrangement characteristics of the superconducting sub-cable. This achieves integrated and simultaneous analysis of macroscopic bending hysteresis behavior and microscopic slip contact field, enabling the solution of the complete mechanical evolution law of the sub-cable under electromagnetic-tension-bending composite loads. This invention clarifies the multi-parameter control mechanism of electromagnetic load, helical lay-up angle, pre-tension strain, and Coulomb friction coefficient on the overall bending stiffness, viscosity-slip transition critical curvature, and local slip stress and contact equivalent stress distribution of the sub-cable. It quantitatively reveals the synergistic evolution law of macroscopic hysteretic deformation and interfacial friction slip behavior under the coupled action of electromagnetic force and mechanical load, and elucidates the key influences of interfacial contact compression and frictional dissipation on the cyclic bending service performance of the superconducting sub-cable. This invention improves computational efficiency while ensuring computational accuracy and fully preserving the microscopic information of strand slip stress, contact stress, and cyclic friction energy consumption. It establishes a complete evaluation system for the mechanical properties of superconducting sub-cables, covering macroscopic bending stiffness, critical slip curvature, cyclic friction energy consumption, and local contact stress. It also improves the theoretical system for multi-field coupling analysis of stick-slip bending of multi-stage superconducting helical sub-cables under strong electromagnetic service conditions.

[0038] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A quantitative analysis method for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading, characterized in that, Includes the following steps: S1. Establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. S2. Based on the aforementioned thin rod equilibrium equation, and combined with the fourth-order Runge-Kutta numerical propulsion strategy, solve the distribution of internal forces and contact forces of the strand within the entire arc length. Determine the viscous-slip bending state at each location based on the relationship between viscous stress and slip stress, and calculate the correspondence between the total bending moment of the sub-cable and the current bending curvature based on the determination results. S3. During the unloading phase, based on the determination function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, the inflection point of the viscous-slip state during the unloading process is determined, the bending moment-curvature curve of the unloading section is obtained, and then the energy dissipation per unit length of sub-cable during the loading and unloading cycle is calculated.

2. The method according to claim 1, characterized in that, The process of establishing the equilibrium equations for the thin rod includes: in the Frenet local coordinate system, using the arc coordinates of the strands as variables, establishing the equilibrium relationship between the normal curvature, subnormal curvature, and tangential torsion of the deformed strands and the internal forces and internal moments distributed along the arc length of the strands; and introducing an electromagnetic force load term and a Coulomb friction coefficient term into this equilibrium relationship. The electromagnetic force load term is expressed as the product of current and magnetic field, and the Coulomb friction coefficient term is used to express the proportional relationship between the contact force of the normal and subnormal lines and the tangential friction force.

3. The method according to claim 2, characterized in that, The process of introducing boundary conditions under the initial pre-tension state includes: setting the initial tangential strain of the strand and the initial normal contact force caused by tension; substituting the cross-sectional area of ​​the strand as a parameter; solving the internal force and internal moment components at each arc length position under the uniaxial tension state along the arc length of the strand; and using the internal force and contact force values ​​at the neutral axis position as the initial values ​​for solving the bending deformation.

4. The method according to claim 1, characterized in that, The process for determining the viscous-slip bending state includes: obtaining the slip stress based on the tangential strain caused by friction; using the relationship between the distance from the center line of the strand to the neutral axis of the sub-cable under the current bending curvature and the viscous stress and slip stress as the basis for determination; using the critical bending curvature as the dividing threshold; when the stress on the strand is less than or equal to the stress value corresponding to the critical bending curvature, it is determined to be in a viscous state; otherwise, it is determined to be in a slip state; and the bending stiffness of the strand is calculated under both states.

5. The method according to claim 4, characterized in that, The calculation process of the correspondence between the total bending moment of the sub-cable and the current bending curvature includes: dividing the winding angle into several discrete segments within a set angle range, calculating the bending stiffness of each discrete segment in the viscous state and the bending stiffness in the slip state, projecting and summing the bending stiffness of each discrete segment according to the symmetry of the first-order helix, and then multiplying it by the current bending curvature to obtain an explicit expression of the total bending moment of the sub-cable.

6. The method according to claim 1, characterized in that, The process of constructing the determination function includes: assuming that the sub-cable is in a completely viscous state during unloading, calculating the sub-cable bending moment corresponding to the cumulative unloading curvature, comparing this bending moment with the peak coordinate of the bending moment-curvature curve, determining that the state is viscous when the unloading bending moment in the assumed completely viscous state is less than the peak bending moment, and determining that the state has entered a stick-slip state when the unloading bending moment in the assumed completely viscous state is greater than or equal to the peak bending moment, and using the bending curvature corresponding to the inflection point of entering the stick-slip state as the critical curvature of the unloading stage; the process of calculating the dissipated energy includes: integrating and subtracting the loading curve and the unloading curve within the bending curvature interval to obtain the dissipated energy per unit length of sub-cable in a complete loading and unloading cycle.

7. A quantitative analysis system for the viscous-slip bending behavior of a superconducting sub-cable under electromagnetic force loading, characterized in that, The system for implementing the method of any one of claims 1-6 comprises: The initial condition setting module is used to establish the equilibrium equation of a single strand in a superconducting sub-cable under electromagnetic force load, and introduce the boundary conditions under the initial pre-tension state as the initial conditions for subsequent bending deformation solution. The stick-slip bending calculation module is loaded to solve the distribution of internal forces and contact forces of the strand over the entire arc length based on the balance equation of the thin rod and combined with the fourth-order Runge-Kutta numerical propulsion strategy. The stick-slip bending state at each position is determined according to the relationship between viscous stress and slip stress, and the correspondence between the total bending moment of the sub-cable and the current bending curvature is calculated based on the determination result. The unloading energy consumption calculation module is used to determine the inflection point of the viscous-slip state during the unloading phase based on the judgment function of the sub-cable bending moment and peak coordinate corresponding to the cumulative unloading curvature under the assumption of complete viscous state, obtain the bending moment-curvature curve of the unloading section, and then calculate the energy dissipation per unit length of sub-cable during the loading and unloading cycle.

8. A computer terminal device, characterized in that, include: One or more processors; A memory, coupled to the processor, for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors perform the steps of the method as described in any one of claims 1-6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1-6.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1-6.

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