Aero-engine transmission part damage analysis method based on contact dislocation coupling

Abstract

The invention discloses an aero-engine transmission part damage analysis method based on contact dislocation coupling, and relates to the technical field of damage analysis, and the aero-engine transmission part damage analysis method comprises the following steps: S1, dividing a raceway surface in contact with a ball of a transmission part into a plurality of uniform cubic units, and constructing a first control equation and a second control equation; s2, obtaining contact pressure distribution of the balls on the inner surface of the raceway; s3, determining free energy; s4, determining a control equation of the coating and the precipitated phase; s5, according to the free energy and the control equation of the coating and the precipitated phase, iterative processing is carried out until the state of the dislocation and the state of the precipitated phase are balanced; and S6, generating a damage analysis result after the states of the dislocation and the precipitated phase are balanced. According to the method, a parameterization tool is provided for damage analysis of the aero-engine in a complex working environment, so that accurate prediction and maintenance are realized.

Description

Technical Field

[0001] This invention relates to the field of damage analysis technology, and specifically to a damage analysis method for aero-engine transmission components based on contact dislocation coupling. Background Technology

[0002] As the heart of an aircraft, the aero-engine is a core component of the modern aircraft power system. Its transmission system (such as main bearings and auxiliary bearings) plays a crucial role in supporting the rotor system, transmitting loads, and ensuring power transmission under extreme operating conditions. However, due to the special operating environment of aero-engines, transmission components, especially bearings, are constantly challenged by high temperatures, high speeds, heavy loads, and complex dynamic loads, making them highly susceptible to performance degradation or sudden failure due to harsh operating conditions.

[0003] From a macroscopic failure mechanism perspective, external loads such as impacts and overloads can cause dangerous situations in transmission components, including rotor drop, motor overload, and heavy-load impacts. Furthermore, considering the dynamic meshing process between the engine's rotating components and bearing assemblies, various conditions can lead to significant mechanical wear. The contact interfaces are subjected to complex localized high temperatures, overload stress, and cyclic stress. Localized high temperatures can severely damage the structural integrity of the coating; the sheer volume of stress generated by cyclic contact leads to the accumulation of plastic deformation on the surface of the transmission components over time, resulting in significant strain hardening. Ultimately, this causes coating peeling and irreversible changes in surface geometry, accelerating the performance degradation of the transmission system.

[0004] From a microstructural perspective, bearing surfaces exhibit multi-scale failure characteristics under extreme environments. Due to high contact stress, phase structure changes and dislocation proliferation or annihilation occur in the plastic deformation zone, resulting in high-density dislocation aggregation. This formation of dislocation loops simultaneously leads to plasticity, significantly enhancing the local stress concentration effect. Microcracks arise around precipitates or inclusions due to strain incompatibility. This type of micro-damage can lead to increased roughness of transmission components, thermal protection failure, and other hazards. Without timely prediction, the resulting non-containment failures can significantly reduce aircraft reliability and create dangerous hazards.

[0005] In light of this, revealing the contact dislocation coupling damage law of engine transmission systems under high temperature, high speed, and high load conditions is a crucial research path for overcoming the bottleneck of long-life design in aero-engines and achieving cost optimization throughout the entire equipment life cycle. Currently, damage analysis of aero-engine transmission components still faces many limitations. These components are subjected to multi-field coupling effects of contact, heat, force, and microstructure in their working environment, resulting in significant nonlinear characteristics in the failure process. However, for numerical calculations, existing finite element method (FEM) software can only handle heat-force coupling problems and lacks a comprehensive model capable of handling complex contact-heat-force-microstructure multi-field coupling. Furthermore, even for simple dynamic heat-force coupling problems, FEM cannot achieve a balance between accuracy and efficiency, and may face convergence difficulties, failing to effectively handle dynamic cyclic evolution models of microstructures. In addition, the traditional continuum mechanics framework, by treating materials as homogeneous isotropic bodies, struggles to characterize the interaction mechanism between dislocation networks and precipitates, making it difficult to analyze dynamic dislocation evolution and quantify the interaction mechanism between dislocations and precipitates, significantly impacting the accuracy of the analysis. Therefore, to overcome the above difficulties, it is necessary to construct a system that fully considers the multi-field coupling of contact, heat, force, and microstructure, and can reflect the evolution law of dynamic dislocation microstructure, so as to accurately analyze the damage of aero-engine transmission components, thereby achieving rapid calculation and accurate prediction, providing an algorithm for further contact behavior analysis and deep optimization, and promoting the further development of aero-engine technology. Summary of the Invention

[0006] To address the above problems, this invention proposes a damage analysis method for aero-engine transmission components based on contact dislocation coupling.

[0007] The technical solution of this invention is: a damage analysis method for aero-engine transmission components based on contact dislocation coupling, comprising the following steps:

[0008] S1. Divide the raceway surface that contacts the balls of the transmission component into several uniform cubic units, and construct the first control equation and the second control equation.

[0009] S2. Based on the first and second control equations, the contact pressure distribution of the balls on the inner surface of the raceway is obtained.

[0010] S3. Determine the free energy;

[0011] S4. Determine the governing equations for the coating and precipitated phases based on the contact pressure distribution of the balls on the inner surface of the raceway.

[0012] S5. Based on the free energy and the governing equations of the coating and precipitates, perform iterative processing until the states of dislocations and precipitates reach equilibrium.

[0013] S6. After the dislocations and precipitated phases reach equilibrium, damage analysis results are generated.

[0014] Furthermore, in S1, the expression for the first governing equation is:

[0015] ;

[0016] In the formula, (k,l) represents a point located on the contact surface between the ball and the raceway of the transmission component, k represents the abscissa of the point located on the contact surface between the ball and the raceway of the transmission component, l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component, and l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component. s K represents the set of source points for the balls and raceways. i-k,j-l p represents the displacement influence coefficient caused by contact. kl h represents the magnitude of the load distribution at the source point. ij The distance between the ball surface and the auxiliary bearing rolling into the coating when no load is applied, w represents the contact depth, and p represents the contact depth. ij The values ​​represent the pressure distribution at the contact point, where i represents the x-coordinate of the contact point after the load is applied, j represents the y-coordinate of the contact point after the load is applied, and I... c This represents the set of contact points between the balls and the raceways;

[0017] In S1, the expression for the second governing equation is:

[0018] .

[0019] Furthermore, in S2, the contact pressure distribution of the ball on the inner surface of the raceway is obtained based on the contact gap and the external load.

[0020] The relationship between the external load and the contact pressure distribution of the balls on the inner surface of the raceway is expressed as follows:

[0021] ;

[0022] In the formula, I s Let x represent the set of source points of the balls and raceways, x represent the abscissa of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, y represent the ordinate of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, Δx represent the size of the surface cube along the horizontal axis, Δy represent the size of the unit surface cube along the vertical axis, p(x,y) represent the distribution of the contact pressure of the balls on the inner surface of the raceways at (x,y) after the application of external load, and W represent the external load.

[0023] Furthermore, in S3, free energy The calculation formula is:

[0024] ;

[0025] In the formula, E crystal E represents the crystal energy. grad E represents the gradient energy of the precipitated phase at the interface caused by inhomogeneity. pre E represents the elastic energy of the precipitated phase. pre-dis The expression represents the interaction energy between dislocations and the precipitated phase, where W represents the external load and c represents the concentration variable of the precipitated phase. Represents a structure variable.

[0026] Furthermore, in S4, the governing equations for the coating and the precipitated phase are expressed as follows:

[0027] ;

[0028] In the formula, C ijkl (·) represents the elastic modulus of the bearing raceway. S represents the elastic modulus of the precipitated phase. klmn (·) represents the strain Eshelby tensor. This represents the elastic strain parameter based on the contact pressure distribution. Indicates intrinsic strain within the precipitate. This represents the equivalent intrinsic strain caused by the material heterogeneity inside and outside the precipitate. It represents the inelastic strain present in a material, including strain caused by temperature changes, lattice misfits, and dislocation motion.

[0029] Furthermore, S5 includes the following sub-steps:

[0030] S51. Based on the free energy, construct the morphological evolution equation for the governing equations of the coating and the precipitated phase;

[0031] S52. Based on the morphological evolution equation, use the Fourier spectral method to solve the equations for concentration variables and structure variables;

[0032] S53. Construct dislocation evolution equations for the governing equations of coatings and precipitates;

[0033] S54. Determine whether the dislocation has reached equilibrium based on the dislocation evolution equation. If yes, proceed to S56; otherwise, proceed to S55.

[0034] S55. Based on the dislocation evolution equation, use the Fourier spectral method to iteratively solve the equation for dislocation density until the dislocations reach equilibrium, and then proceed to S56.

[0035] S56. Determine the state of the precipitated phase based on the equations for the concentration variable and the structure variable. If the state of the precipitated phase is not in equilibrium, update the time step until the state of the precipitated phase reaches equilibrium.

[0036] Furthermore, in S51, the expression for the morphological evolution equation is:

[0037] ;

[0038] In the formula, Let represent the free energy, and 'c' represent the concentration variable. Let represent structural variables, t represent time, and M represent the dynamic coefficient of grain boundary mobility. This represents the Nabla operator.

[0039] Furthermore, in S53, the expression for the dislocation evolution equation is:

[0040] ;

[0041] In the formula, ρ represents the dislocation density, α represents the slip system number, t represents time, n=1 represents a negative dislocation, and n=2 represents a positive dislocation. Let represent the dislocation velocity, b represent the Burgers vector, and b represent the magnitude of the Burgers vector.

[0042] Furthermore, S6 includes the following sub-steps:

[0043] S61. After the dislocations and precipitated phases reach equilibrium, determine the distribution of the elastic field and temperature field at point (x,y) based on the dislocation density, concentration variable, and structural variable; where x represents the abscissa of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division, and y represents the ordinate of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division.

[0044] S62. Based on the distribution of elastic field and temperature field at point (x,y), the surface deformation results are obtained using the governing equations of coating and precipitated phase.

[0045] S63. Determine whether the surface deformation result meets the requirements. If so, the surface deformation result is used as the damage analysis result; otherwise, the contact morphology is updated, and the process returns to S1 until a converged damage analysis result is obtained. This represents the distance between the ball and the raceway calculated at time step (n+1). ε represents the distance between the ball and the raceway calculated at the nth time step, and ε is an infinitesimally small positive number specifically set to ensure the convergence of the distance result.

[0046] The beneficial effects of this invention are:

[0047] (1) This invention is based on dynamic coupling of contact microstructure for damage analysis, and establishes a model algorithm that integrates contact-thermal-mechanical-microstructure multi-field coupling, which is closer to the complex actual working conditions of engine high temperature and heavy load, and achieves the unity of accuracy and efficiency.

[0048] (2) By assuming equivalent intrinsic strain, the present invention transforms the heterogeneous and nonlinear problem that is difficult to solve directly into a linear superposition problem with the same elastic modulus. In addition, the present invention combines energy control methods and comprehensively utilizes Fourier spectroscopy to quantify the dynamic evolution law of dislocations and precipitates, and accurately analyzes the damage effect of microstructure changes on transmission components.

[0049] (3) This invention provides a parameterized tool for damage analysis of aero-engines under complex working environments, so as to achieve accurate prediction and maintenance. Attached Figure Description

[0050] Figure 1 This is a flowchart of a method for damage analysis of aero-engine transmission components based on contact dislocation coupling. Detailed Implementation

[0051] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0052] like Figure 1 As shown, this invention provides a damage analysis method for aero-engine transmission components based on contact dislocation coupling, comprising the following steps:

[0053] S1. Divide the raceway surface that contacts the balls of the transmission component into several uniform cubic units, and construct the first control equation and the second control equation.

[0054] S2. Based on the first and second control equations, the contact pressure distribution of the balls on the inner surface of the raceway is obtained.

[0055] S3. Determine the free energy;

[0056] S4. Determine the governing equations for the coating and precipitated phases based on the contact pressure distribution of the balls on the inner surface of the raceway.

[0057] S5. Based on the free energy and the governing equations of the coating and precipitates, perform iterative processing until the states of dislocations and precipitates reach equilibrium.

[0058] S6. After the dislocations and precipitated phases reach equilibrium, damage analysis results are generated.

[0059] In this embodiment of the invention, in S1, the expression of the first governing equation is:

[0060] ;

[0061] In the formula, (k,l) represents a point located on the contact surface between the ball and the raceway of the transmission component, k represents the abscissa of the point located on the contact surface between the ball and the raceway of the transmission component, l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component, and l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component.s K represents the set of source points for the balls and raceways. i-k,j-l p represents the displacement influence coefficient caused by contact. kl h represents the magnitude of the load distribution at the source point. ij The distance between the ball surface and the auxiliary bearing rolling into the coating when no load is applied, w represents the contact depth, and p represents the contact depth. ij The values ​​represent the pressure distribution at the contact point, where i represents the x-coordinate of the contact point after the load is applied, j represents the y-coordinate of the contact point after the load is applied, and I... c This represents the set of contact points between the balls and the raceways;

[0062] In S1, the expression for the second governing equation is:

[0063] .

[0064] In this embodiment of the invention, in S2, the contact pressure distribution of the ball on the inner surface of the raceway is obtained based on the contact gap and the external load.

[0065] The relationship between the external load and the contact pressure distribution of the balls on the inner surface of the raceway is expressed as follows:

[0066] ;

[0067] In the formula, I s Let x represent the set of source points of the balls and raceways, x represent the abscissa of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, y represent the ordinate of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, Δx represent the size of the surface cube along the horizontal axis, Δy represent the size of the unit surface cube along the vertical axis, p(x,y) represent the distribution of the contact pressure of the balls on the inner surface of the raceways at (x,y) after the application of external load, and W represent the external load.

[0068] In this embodiment of the invention, in S3, the free energy The calculation formula is:

[0069] ;

[0070] In the formula, E crystal E represents the crystal energy. grad E represents the gradient energy of the precipitated phase at the interface caused by inhomogeneity. pre E represents the elastic energy of the precipitated phase. pre-dis The expression represents the interaction energy between dislocations and the precipitated phase, where W represents the external load and c represents the concentration variable of the precipitated phase. Represents a structure variable.

[0071] E crystalE represents the crystal energy, which can be calculated from the phase diagram of a specific material; grad E represents the gradient energy of the precipitated phase at the interface caused by inhomogeneity; crystal +E grad Part of it is the chemical energy of the material, which is closely related to the concentration and structural morphology of the precipitates; E pre +E pre-dis This is partly referred to as elastic energy.

[0072] In this embodiment of the invention, in S4, the governing equations for the coating and the precipitated phase are expressed as follows:

[0073] ;

[0074] In the formula, C ijkl (·) represents the elastic modulus of the bearing raceway. S represents the elastic modulus of the precipitated phase. klmn (·) represents the strain Eshelby tensor. This represents the elastic strain parameter based on the contact pressure distribution. Indicates intrinsic strain within the precipitate. This represents the equivalent intrinsic strain caused by the material heterogeneity inside and outside the precipitate. It represents the inelastic strain present in a material, including strain caused by temperature changes, lattice misfits, and dislocation motion.

[0075] In this embodiment of the invention, S5 includes the following sub-steps:

[0076] S51. Based on the free energy, construct the morphological evolution equation for the governing equations of the coating and the precipitated phase;

[0077] S52. Based on the morphological evolution equation, use the Fourier spectral method to solve the equations for concentration variables and structure variables;

[0078] S53. Construct dislocation evolution equations for the governing equations of coatings and precipitates;

[0079] S54. Determine whether the dislocation has reached equilibrium based on the dislocation evolution equation. If yes, proceed to S56; otherwise, proceed to S55.

[0080] S55. Based on the dislocation evolution equation, use the Fourier spectral method to iteratively solve the equation for dislocation density until the dislocations reach equilibrium, and then proceed to S56.

[0081] S56. Determine the state of the precipitated phase based on the equations for the concentration variable and the structure variable. If the state of the precipitated phase is not in equilibrium, update the time step until the state of the precipitated phase reaches equilibrium.

[0082] In this embodiment of the invention, in S51, the expression for the morphological evolution equation is:

[0083] ;

[0084] In the formula, Let represent the free energy, and 'c' represent the concentration variable. Let represent structural variables, t represent time, and M represent the dynamic coefficient of grain boundary mobility. This represents the Nabla operator.

[0085] In this embodiment of the invention, in S53, the expression for the dislocation evolution equation is:

[0086] ;

[0087] In the formula, ρ represents the dislocation density, α represents the slip system number, t represents time, n=1 represents a negative dislocation, and n=2 represents a positive dislocation. Let represent the dislocation velocity, b represent the Burgers vector, and b represent the magnitude of the Burgers vector.

[0088] In this embodiment of the invention, S6 includes the following sub-steps:

[0089] S61. After the dislocations and precipitated phases reach equilibrium, determine the distribution of the elastic field and temperature field at point (x,y) based on the dislocation density, concentration variable, and structural variable; where x represents the abscissa of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division, and y represents the ordinate of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division.

[0090] S62. Based on the distribution of elastic field and temperature field at point (x,y), the surface deformation results are obtained using the governing equations of coating and precipitated phase.

[0091] S63. Determine whether the surface deformation result meets the requirements. If so, the surface deformation result is used as the damage analysis result; otherwise, the contact morphology is updated, and the process returns to S1 until a converged damage analysis result is obtained. This represents the distance between the ball and the raceway calculated at time step (n+1). ε represents the distance between the ball and the raceway calculated at the nth time step, and ε is an infinitesimally small positive number specifically set to ensure the convergence of the distance result.

[0092] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A damage analysis method for aero-engine transmission components based on contact dislocation coupling, characterized in that, Includes the following steps: S1. Divide the raceway surface that contacts the balls of the transmission component into several uniform cubic units, and construct the first control equation and the second control equation. S2. Based on the first and second control equations, the contact pressure distribution of the balls on the inner surface of the raceway is obtained. S3. Determine the free energy; S4. Determine the governing equations for the coating and precipitated phases based on the contact pressure distribution of the balls on the inner surface of the raceway. S5. Based on the free energy and the governing equations of the coating and precipitates, perform iterative processing until the states of dislocations and precipitates reach equilibrium. S6. After the dislocations and precipitated phases reach equilibrium, damage analysis results are generated.

2. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, In S1, the expression for the first governing equation is: ； In the formula, (k,l) represents a point located on the contact surface between the ball and the raceway of the transmission component, k represents the abscissa of the point located on the contact surface between the ball and the raceway of the transmission component, l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component, and l represents the ordinate of the point located on the contact surface between the ball and the raceway of the transmission component. s K represents the set of source points for the balls and raceways. i-k,j-l p represents the displacement influence coefficient caused by contact. kl h represents the magnitude of the load distribution at the source point. ij The distance between the ball surface and the auxiliary bearing rolling into the coating when no load is applied, w represents the contact depth, and p represents the contact depth. ij The values ​​represent the pressure distribution at the contact point, where i represents the x-coordinate of the contact point after the load is applied, j represents the y-coordinate of the contact point after the load is applied, and I... c This represents the set of contact points between the balls and the raceways; In S1, the expression for the second governing equation is: 。 3. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, In S2, the contact pressure distribution of the ball on the inner surface of the raceway is obtained based on the contact gap and the external load. The relationship between the external load and the contact pressure distribution of the balls on the inner surface of the raceway is expressed as follows: ； In the formula, I s Let x represent the set of source points of the balls and raceways, x represent the abscissa of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, y represent the ordinate of the source points in the Cartesian coordinate system on the horizontal plane after surface element division, Δx represent the size of the surface cube along the horizontal axis, Δy represent the size of the unit surface cube along the vertical axis, p(x,y) represent the distribution of the contact pressure of the balls on the inner surface of the raceways at (x,y) after the external load is applied, and W represent the external load.

4. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, In S3, the free energy The calculation formula is: ； In the formula, E crystal E represents the crystal energy. grad E represents the gradient energy of the precipitated phase at the interface caused by inhomogeneity. pre E represents the elastic energy of the precipitated phase. pre-dis The expression represents the interaction energy between dislocations and the precipitated phase, where W represents the external load and c represents the concentration variable of the precipitated phase. Represents a structure variable.

5. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, In S4, the governing equations for the coating and the precipitated phase are expressed as follows: ； In the formula, C ijkl (·) represents the elastic modulus of the bearing raceway. S represents the elastic modulus of the precipitated phase. klmn (·) represents the strain Eshelby tensor. This represents the elastic strain parameter based on the contact pressure distribution. Indicates intrinsic strain within the precipitate. This represents the equivalent intrinsic strain caused by the material heterogeneity inside and outside the precipitate. This represents the inelastic strain present in the material.

6. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, S5 includes the following sub-steps: S51. Based on the free energy, construct the morphological evolution equation for the governing equations of the coating and the precipitated phase; S52. Based on the morphological evolution equation, use the Fourier spectral method to solve the equations for concentration variables and structure variables; S53. Construct dislocation evolution equations for the governing equations of coatings and precipitates; S54. Determine whether the dislocation has reached equilibrium based on the dislocation evolution equation. If yes, proceed to S56; otherwise, proceed to S55. S55. Based on the dislocation evolution equation, use the Fourier spectral method to iteratively solve the equation for dislocation density until the dislocations reach equilibrium, and then proceed to S56. S56. Determine the state of the precipitated phase based on the equations for the concentration variable and the structure variable. If the state of the precipitated phase is not in equilibrium, update the time step until the state of the precipitated phase reaches equilibrium.

7. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 6, characterized in that, In S51, the expression for the morphological evolution equation is: ； In the formula, Let represent the free energy, and 'c' represent the concentration variable. Let represent structural variables, t represent time, and M represent the dynamic coefficient of grain boundary mobility. This represents the Nabla operator.

8. The damage analysis method for aero-engine transmission components based on contact dislocation coupling according to claim 6, characterized in that, In S53, the expression for the dislocation evolution equation is: ； In the formula, ρ represents the dislocation density, α represents the slip system number, t represents time, n=1 represents a negative dislocation, and n=2 represents a positive dislocation. Let represent the dislocation velocity, b represent the Burgers vector, and b represent the magnitude of the Burgers vector.

9. The method for damage analysis of aero-engine transmission components based on contact dislocation coupling according to claim 1, characterized in that, S6 includes the following sub-steps: S61. After the dislocations and precipitated phases reach equilibrium, determine the distribution of the elastic field and temperature field at point (x,y) based on the dislocation density, concentration variable, and structural variable; where x represents the abscissa of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division, and y represents the ordinate of the source point on the horizontal plane in a Cartesian coordinate system after the surface unit division. S62. Based on the distribution of elastic field and temperature field at point (x,y), the surface deformation results are obtained using the governing equations of coating and precipitated phase. S63. Determine whether the surface deformation result meets the requirements. If so, the surface deformation result is used as the damage analysis result; otherwise, the contact morphology is updated, and the process returns to S1 until a converged damage analysis result is obtained. This represents the distance between the ball and the raceway calculated at time step (n+1). ε represents the distance between the ball and the raceway calculated at the nth time step, and ε is an infinitesimally small positive number specifically set to ensure the convergence of the distance result.

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