A spline curve-based spline topology optimization method

Abstract

The application relates to the technical field of aviation transmission components, and discloses a spline topology optimization method based on a B-spline curve, which takes five-node cubic B-spline theory as the core, first clearly defines a spline tooth surface modification interval, then sets key control parameters such as a left end modification amount, a left end middle modification parameter, a tangent point position parameter, a right end middle modification parameter and a right end modification amount; subsequently, a parameter collaborative constraint mechanism is established, the numerical interval of the left end middle modification parameter is locked by relying on the left end modification amount, the right end modification amount and the tangent point position parameter, and the specific value of the right end middle modification parameter is derived; then all the given modification parameters and interpolation base functions are substituted into a general expression of the B-spline curve to generate a final modification curve equation; finally, a complete modification tooth surface is projected and formed according to the equation, a B-spline track machining scheme is planned and cutting parameters are set, machining assembly errors are adapted through parameterized design, simulation optimization and precision detection are combined, and it is ensured that the tooth surface precision meets the use requirements.

Description

Technical Field

[0001] This invention relates to the field of aerospace transmission component technology, specifically a spline topology optimization method based on B-spline curves. It is applicable to spline connection scenarios under high speed and heavy load conditions, and can effectively improve the contact reliability and service life of spline assembly. It is especially suitable for application scenarios in aerospace transmission systems with stringent requirements for tooth surface contact accuracy and anti-error disturbance capability. Background Technology

[0002] In aerospace transmission systems, splines, as core power transmission components, must withstand high speeds, large torques, and severe vibrations under complex operating conditions. The contact state of their tooth surfaces directly determines transmission accuracy, system stability, and service life. Existing spline modification techniques often employ simplified models such as parabolic and straight lines to design tooth surfaces, which has the following key drawbacks: 1. The parametric adjustment of the profile curve lacks flexibility and cannot be accurately adapted to the specific specifications and working conditions of non-standard splines. The coordination and matching of profiles of different sections are poor. 2. Lack of global continuity constraints makes the tooth surface prone to structural defects such as abrupt tangent changes, depressions, or pits, leading to stress concentration after assembly and exacerbating tooth surface wear and fatigue damage. 3. It has weak resistance to disturbances caused by equipment errors such as tooth thickness and pitch angle. The simulation results deviate significantly from the actual assembly conditions, making it difficult to meet the high precision requirements of aerospace transmission. 4. Existing shaping adjustments mostly rely on empirical machining parameter adjustments or manual shaping, which is a complex process that requires repeated measurements, resulting in low efficiency, poor consistency, and insufficient accuracy in machining path optimization.

[0003] As mentioned in the published patent CN119598821A, the core requirement for spline reshaping is to reduce contact stress and improve assembly adaptability through precise structural design and processing control. However, the existing technology has not yet achieved a deep integration of reshaping structure with error adaptation and stress optimization. Summary of the Invention

[0004] (a) Technical problems to be solved To address the shortcomings of existing technologies, this invention provides a spline topology optimization method based on B-spline curves, which effectively alleviates stress concentration and reduces the risk of tooth surface wear. It solves the problem of large deviations between simulation and actual assembly in existing technologies. At the same time, through parametric control and special tool processing, it avoids the reliance on experience in manual shaping, improves the consistency of shaping effect and production efficiency, and significantly enhances the reliability and safety of aerospace transmission systems.

[0005] (II) Technical Solution To achieve the above objectives, the present invention provides the following technical solution: a spline topology optimization method based on B-spline curves, comprising the following steps: Step 1: Determine the spline tooth surface modification interval based on the five-node cubic B-spline theory; Step 2: Set the key control parameters for the shaping curve, including the left end shaping amount. Left end middle shaping parameters Tangent point position parameters Right end middle section shaping parameters Right end shaping amount ; Step 3: Establish a parameter coordination constraint mechanism and utilize... , , Lock left-side center shaping parameters The numerical range is then used to derive the shaping parameters for the middle right end through formulas. The value; Step 4: Based on all the determined shaping parameters and interpolation basis functions, substitute them into the general expression for B-spline curves to obtain the final shaping curve equation; Step 5: Based on the final modified curve equation, project and form the complete modified tooth surface, plan the B-spline trajectory machining scheme and set the cutting parameters, adapt the machining and assembly errors through parametric design, and combine simulation optimization and detection to ensure that the tooth surface accuracy meets the standards.

[0006] Preferably, in step one, the shaping interval is set along the tooth width direction, and the tooth width direction is defined as the stretching coordinate. The reshaping range is ,in, It represents the width of the spline teeth.

[0007] Preferably, in step two, represent The amount of reshaping at the location, represent The amount of reshaping at the location, The coordinates of the point of tangency between the profile curve and the tip line of the spline tooth in the tooth width direction are represented, with a value range of [value missing]. .

[0008] Preferably, in step three, the shaping parameters of the left-end center are locked. The formula for the numerical range is expressed as follows: ; in, represent The lower limit of the value; represent The upper limit of the possible values; This means that the constraint conditions apply to all discrete measurement points in the tooth width direction; Represents the range of reshaping intervals The first A discrete sampling point; Representing the The coefficient terms derived from the basis functions of each discrete sampling point; Representing the A linear combination coefficient term of discrete sampling points combined with the shaping amounts at both ends.

[0009] Preferably, the coefficient terms derived from the basis functions The calculation formula is: ; in, Represents corresponding The cubic B-spline basis function values; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; represent At the tangent point The cubic B-spline basis function value at the location.

[0010] Preferably, the linear combination coefficient term combining the shaping amounts at both ends... The calculation formula is: ; in, Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at the location.

[0011] Preferably, the formula for calculating the cubic B-spline basis function value is: ; ; ; in, represent Number B-spline basis functions; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; Representative Table Number B-spline basis function values; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; represent Number The basis function value of the B-spline and Together they constitute the recursive computation terms of higher-order basis functions.

[0012] Preferably, the B-spline basis functions are given in Cox-deBoor recursive form, wherein the zero-degree basis function is defined as follows: ; in, The parameter variables represent the B-spline curve.

[0013] Preferably, in step three, the shaping parameters of the right-end middle section are derived. The numerical derivation formula is as follows: ; Preferably, in step four, the formula expression for the final modified curve equation is: ; in, Represents the position in the tooth width direction The corresponding shaping amount value; Representing the The shaping parameters corresponding to each control point.

[0014] Compared with existing technologies, this invention provides a spline topology optimization method based on B-spline curves, which has the following beneficial effects: 1. The present invention significantly reduces contact stress: The convex continuous tooth surface structure design enables the contact area to be evenly distributed after spline assembly. Even under working conditions with slight angular misalignment, it can effectively disperse the stress on the tooth surface, greatly alleviate the stress concentration problem that is common in traditional shaping methods, reduce tooth surface wear from the root, and improve the tooth surface load-bearing stability. Compared with the existing helical angle shaping technology, the stress dispersion effect is better, further reducing the risk of tooth surface fatigue damage.

[0015] 2. Improved assembly adaptability and anti-interference capability of the present invention: Through the continuous design of tooth surface tangent, the coordinated matching of different cross-sectional modification dimensions is ensured, the spline's adaptability to assembly deviation is enhanced, and it can stably withstand a certain range of angular misalignment and tooth thickness error. At the same time, it keeps the assembly gap in a uniform state, effectively reducing the deviation between simulation design and actual assembly application, and improving assembly accuracy and transmission stability.

[0016] This invention optimizes processing consistency and efficiency: relying on a parametric control system combined with specialized tool processing technology, it eliminates the dependence on operator experience in traditional manual shaping, ensures the uniformity of spline shaping effects in mass production, simplifies the processing flow, improves production efficiency, and also has good flexibility, which can be directly adapted to the customized needs of non-standard splines, thus broadening the application scenarios.

[0017] 4. The service life of this invention is greatly extended: The optimized tooth surface structure can effectively reduce local excessive wear and fatigue damage, improve the overall structural strength and wear resistance of the spline. Actual installation verification shows that compared with traditional splines, the service life is significantly extended, and it can stably achieve long-term continuous fault-free operation, further enhancing the reliability and operational safety of the aviation transmission system. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the shaping curve of the present invention; Figure 2 This is a comparison diagram of the contact stress of the external spline before and after the modification of the present invention; Figure 3 This is a diagram illustrating the implementation steps of the method of the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. It is worth noting that this application also relates to prior art. Since prior art is well known to those skilled in the art, it will not be described in detail in this application.

[0020] Please see Figures 1-3 A spline topology optimization method based on B-spline curves includes the following steps: Step 1: Based on the five-node cubic B-spline theory, determine the spline tooth surface modification interval. The modification interval is set along the tooth width direction, and the tooth width direction is defined as the extrusion coordinate. The reshaping range is ,in, Represents the width of the spline teeth; Step one employs five-node cubic B-spline theory as the basis for constructing the shaping interval and tooth surface, balancing the local controllability and overall continuity of the curve. Compared to traditional linear or parabolic shaping methods, it can achieve refined and abrupt tooth surface shaping by adjusting control points, effectively avoiding local stress concentration problems caused by uneven tooth surface transitions under high-speed and heavy-load conditions. Simultaneously, the shaping interval is clearly defined in the tooth width direction. Within the range, it can accurately match the actual force contact area of ​​the spline, ensuring that the shaping effect fully covers the core working section. This does not increase the machining difficulty, but maximizes the contact uniformity and load-bearing capacity of the tooth surface, providing a clear and controllable basic framework for subsequent parameter constraints and machining implementation. Step 2: Set the key control parameters for the shaping curve. The key control parameters for the shaping curve include the shaping amount at the left end. Left end middle shaping parameters Tangent point position parameters Right end middle section shaping parameters Right end shaping amount , represent The amount of reshaping at the location, represent The amount of reshaping at the location, The coordinates of the point of tangency between the profile curve and the tip line of the spline tooth in the tooth width direction are represented, with a value range of [value missing]. ; Step 2 involves adjusting the shaping amount ∆1 at both ends. The profile modification range is directly anchored to the tooth width boundary to ensure that the boundary area adapts to the assembly clearance requirements. The profile modification parameters are located in the middle of the left end. Right end middle section shaping parameters The surface morphology of the middle section of the tooth width is specifically optimized to avoid contact stress concentration in the middle region, and the tangent point position parameters are adjusted accordingly. As the core constraint point, it is limited to the tooth width. arrive The range of values ​​ensures the continuity of the tangent between the shaping curve and the tooth tip line, eliminating abrupt changes in cross-section transition, and also serves as a key basis for parameter coordination constraints, thus providing a basis for subsequently locking the shaping parameters of the middle left end. Interval, derivation of the right-hand middle shaping parameters It provides a clear geometric reference, covering the entire tooth width range, taking into account both the flexibility of local adjustment and the stability of the global structure, adapting to the use requirements of uniform tooth surface contact under high speed and heavy load conditions, and realizing the unity of precise segmented control of the profile curve and global smooth constraint. Step 3: Establish a parameter coordination constraint mechanism and utilize... , , Lock left-side center shaping parameters The numerical range of the given value is expressed by the formula: ; in, represent The lower limit of the value; represent The upper limit of the possible values; This means that the constraint conditions apply to all discrete measurement points in the tooth width direction; Represents the range of reshaping intervals The first A discrete sampling point; Representing the The coefficient terms derived from the basis functions of each discrete sampling point; Representing the A linear combination coefficient term of discrete sampling points combined with the shaping amounts at both ends; By transforming the global convexity constraint and tangent continuity constraint of the shaping curve into a quantifiable parameter value range, the shaping parameters of the left-side middle section are adjusted. The formula achieves precise and efficient limitation by using discrete measurement points covering the entire tooth width range. Constraint calculations are performed to avoid local failures caused by single-location constraints, ensuring... The value of can meet the convexity requirements of the curve throughout the entire modification range, eliminating defects such as depressions and pits on the tooth surface from the root. At the same time, the formula is directly related to the modification amount at both ends. , and the location of the tangent point By using basis functions to derive coefficient terms and linear combination coefficient terms It can quickly lock the location without complicated manual iterative calculations. The reasonable range of values ​​greatly improves the efficiency and accuracy of parameter design, provides a quantitative basis for the establishment of parameter coordination constraint mechanism, ensures the consistency and stability of the profile curve shape, and ultimately achieves uniform distribution of tooth surface contact stress to meet the requirements of high speed and heavy load. Coefficient terms derived from basis functions The calculation formula is: ; in, Represents corresponding The cubic B-spline basis function values; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; represent At the tangent point The cubic B-spline basis function value at point; formula pass and By directly associating the basis function values ​​corresponding to the two core profile modification parameters at the middle of the left and right ends, the formula accurately captures the difference in influence of the two control points in the middle section of the tooth width on the profile modification curve morphology. Simultaneously, the formula incorporates the tangent point position. corresponding , This transforms the geometric requirements of tangent continuity constraints into calculable numerical coefficients, avoiding the drawbacks of relying solely on experience to judge curve smoothness in traditional shape modification design, and ensuring the accuracy of the calculated results. The selected values ​​not only meet the requirements of curve convexity but also match the geometric condition of tangent continuity, providing a rigorous numerical basis for the establishment of the entire parameter coordination constraint mechanism. The linear combination coefficient term combining the shaping amounts at both ends The calculation formula is: ; in, Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; formula Directly use the known modification amounts at both ends of the tooth width , As a core variable, it is combined with the basis function values ​​at the corresponding positions. , The boundary shaping requirements are transferred to the middle section of the tooth width to ensure a smooth transition between the middle shaping curve and the two boundary ends, avoiding surface discontinuity issues caused by segmented shaping. Simultaneously, the formula incorporates the tangent point location. corresponding , , This transforms the geometric constraint of tangent continuity into a computable numerical value, enabling... The value of matches both the shaping amplitude at both ends and meets the requirement of a smooth transition at the tangent point, and then is combined with the coefficient term. Cooperation, together constitute The constraint basis of the value range, through quantitative calculation to replace empirical trial calculation, greatly improves the accuracy and efficiency of parameter design, and ultimately ensures the global consistency of the entire profile curve, and achieves uniform distribution of tooth surface contact stress. The formula for calculating the basis function value of a cubic B-spline is: ; ; ; in, represent Number B-spline basis functions; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; Representative Table Number B-spline basis function values; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; represent Number The basis function value of the B-spline and Together they constitute the recursive computation terms of higher-order basis functions; B-spline basis functions are given in Cox-deBoor recursive form, where the zero-degree basis function is defined as follows: ; in, The parameter variables representing the B-spline curve; The cubic B-spline basis function values ​​are recursively derived starting from the zero-order basis function. The basic value rules of the basis function are clarified through piecewise definition, which lays a unified logical benchmark for the iterative calculation of subsequent higher-order basis functions. Then, the secondary basis functions are derived in the form of Cox-deBoor recursion. Higher-order basis functions are generated by linear combination of lower-order basis functions. This not only ensures the local support characteristics of the basis functions, but also accurately guarantees the smoothness and continuity of the cubic B-spline curve, which meets the core requirement of no abrupt transition in tooth surface modification. Then, the shaping parameters of the middle part on the right end are derived using formulas. The numerical value is derived from the following formula: ; The formula directly uses the modification amount at both ends of the tooth width. , And the left-middle parameter of the locked interval As input variables, combined with the parameters at the tangent point position basis functions at , , , The solution can be obtained quickly through simple linear combination and normalization calculation. The numerical values ​​eliminate the need for complex iterative calculations, significantly improving the efficiency and accuracy of parameter design, while deeply integrating tangent continuity constraints and global convexity constraints. In the computational logic, ensure that the solution is Not only can with , , A smooth transition ensures the consistency of the entire profile curve, avoiding stress concentration on the tooth surface caused by discontinuity in the middle parameters. Furthermore, this derivation logic is consistent with the previous... The constraint formulas echo each other, constructing a complete parameter coordination system from boundary parameters to intermediate parameters to global constraints, providing reliable parameter support for the subsequent generation of the final modified curve equation; Step 4: Based on all the determined shaping parameters and interpolation basis functions, substitute them into the general expression for B-spline curves to obtain the final shaping curve equation. The formula is as follows: ;' in, Represents the position in the tooth width direction The corresponding shaping amount value; Representing the The shaping parameters corresponding to each control point; The final shaping curve equation is directly related to all the preceding parameter design stages. It is the key transformation from theoretical parameters to the actual shape of the tooth surface. It not only provides a clear mathematical basis for the subsequent tooth surface projection forming, but also lays a precise model foundation for machining trajectory planning and simulation verification, ensuring the consistency and reliability of the technical solution from design to implementation. Step 5: Based on the final modified curve equation, project and form the complete modified tooth surface, plan the B-spline trajectory machining scheme and set the cutting parameters, adapt the machining and assembly errors through parametric design, and combine simulation optimization and inspection to ensure that the tooth surface accuracy meets the standards. Specifically, this includes: Based on the determined profile modification curve parameters, the complete modified tooth surface is projected from the tooth tip to the tooth root. In the machining process, a CNC milling machine with a dedicated modification tool is used. Before machining, a spline modification model is generated using 3D modeling software such as UG. After Hypermesh mesh generation and preprocessing, the tooth surface contact stress distribution is simulated and verified using Abaqus finite element software, and machining parameters are optimized. The tool moves along a preset B-spline curve trajectory, with the cutting speed controlled at 30-50 m / min and the feed rate at 0.02-0.05 mm / r, ensuring a tooth surface accuracy of IT6 grade. Simultaneously, parametric design is used to adapt to error disturbances within the tooth thickness ±0.008 mm and pitch angle ±0.003° range. During assembly, a coordinate measuring machine is used to detect the modification amount at each measuring point on the tooth surface, strictly controlling the clearance fluctuation range within 0.003-0.012 mm to ensure machining and assembly reliability.

[0021] Example 1 This embodiment focuses on the transmission spline of a certain type of aero-engine, employing a five-node cubic B-spline modification technique to optimize the tooth surface structure. This addresses issues such as stress concentration and insufficient precision in traditional modification methods, meeting the stringent operating conditions of aero-engine transmissions. The specific implementation process is as follows: I. Target Audience and Basic Parameters This embodiment is adapted to a certain type of aero-engine transmission spline. This spline undertakes high-speed, heavy-load transmission tasks, requiring extremely high uniformity of tooth surface contact, wear resistance, and assembly precision. The core basic parameters are as follows: tooth width L=50mm, made of 40CrNiMoA alloy structural steel, with the tooth surface hardness controlled at HRC38-42 after heat treatment, the core contact area width of the tooth surface is 8mm, the design requirement is a tooth surface precision of IT6 grade, and the assembly clearance fluctuation range is controlled within 0.003-0.012mm. Specific implementation steps

[0022] Step 1: Determine the reshaping area Based on the five-node cubic B-spline theory, the spline tooth surface modification interval is defined, and the stretching coordinates are set along the tooth width direction. The range of the reshaping interval is clearly defined as follows: ,in Corresponding to the left end of the spline, Corresponding to the right end of the spline, it precisely covers the core contact area of ​​the tooth surface and the entire tooth width range, providing a basic framework for subsequent parameter design and curve modeling.

[0023] Step 2: Set key control parameters Set key control parameters for the shaping curve, including the shaping amount at the left end. Left end middle shaping parameters Tangent point position parameters Right end middle section shaping parameters Right end shaping amount Based on the operating requirements of this aerospace spline, determine , , Ensure that the profile curve is tangent to the tooth tip line to meet the requirements for a smooth transition.

[0024] Step 3: Establish a parameter coordination constraint mechanism and solve for the parameters. based on , , Given three parameters, establish a parameter coordination constraint mechanism to lock them. Range of values ​​and derivation The numerical values, and the specific process, are as follows: Determine the degree of the B-spline curve The number of secondary control points is 5. Based on the open interval node vector rules, the node vector is set. It is suitable for the segment uniformity requirement of 50mm tooth width; First, the cubic B-spline basis function values ​​are calculated using the Cox-deBoor recursive form. The zero-degree basis function is calculated using the formula... Higher-order basis functions are defined by recursive formulas. Calculations, obtained synchronously: The corresponding basis function values ​​of each parameter ( , respectively corresponding ); Tangent point position basis functions Calculations yielded , , , ; Secondly, calculate the derivative coefficients of the basis functions. and the linear combination coefficient term combining the shaping amounts at both ends Substitute , , , , , By simplifying the formula and substituting the basis function values ​​of each discrete measuring point, the following results are obtained for the full tooth width at 10 discrete measuring points (0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 mm). and ,For example: hour, ,because If the value is 0, the constraint formula premise is not met, so it is directly excluded. hour, , ; Subsequently, through the constraint formula Seven were selected. The effective measuring points are calculated as follows: The reasonable range of values ​​is Select the optimal value of the interval Substitute into the formula Complete the determination of all shaping parameters.

[0025] Step 4: Generate the final shaping curve equation The already determined , , , and the corresponding cubic B-spline basis functions Substituting into the general expression for B-spline curves, we obtain the final modified curve equation. This equation can accurately output the position of any position in the tooth width direction. shaping amount This provides a clear mathematical basis for tooth surface forming.

[0026] Step 5: Tooth surface forming, machining adaptation, and accuracy verification Based on the final profile curve equation, the complete profiled tooth surface is formed by projecting from the tooth tip to the tooth root. Simultaneously, machining scheme planning, error adaptation, and accuracy testing are carried out. The specific process is as follows: Modeling and simulation optimization: All shaping parameters were imported into UG 3D modeling software to generate a five-node cubic B-spline shaping spline model; the model was imported into Hypermesh software for mesh generation preprocessing, and then Abaqus finite element software was used to simulate the tooth surface contact stress distribution under the 0.009° angular misalignment condition. Based on the simulation results, the machining parameters were optimized and the optimal cutting scheme was determined. CNC machining: A CNC milling machine is used with a special shaping tool. The cutting speed is controlled at 40m / min and the feed rate is 0.03mm / r. The tool moves along the preset B-spline curve trajectory to accurately machine the shaped tooth surface. The error disturbance within the range of tooth thickness ±0.008mm and pitch angle ±0.003° is adapted through parametric design. Precision testing: After machining, 10 measuring points are evenly selected in the z-direction of the tooth width using a coordinate measuring machine to verify that the modification amount of each measuring point meets the design requirements; at the same time, the maximum contact stress on the tooth surface is measured to be 259.5MPa by a pressure sensor, which meets the strength design standard, and the assembly clearance fluctuation range is controlled within 0.005-0.010mm, which meets the preset accuracy requirements.

[0027] III. Application Effects After the modified spline was installed and applied, and after whole-machine debugging and durability testing, the vibration amplitude of the transmission system was reduced by 35% compared with the traditional modified spline, and the operating noise was reduced by 6dB, effectively improving the transmission smoothness. After 2,000 hours of rigorous working condition durability testing, there were no obvious defects such as wear or dents on the tooth surface, the clearance distribution was uniform, and the contact stress was always within a reasonable range, fully meeting the high reliability and high precision transmission requirements of a certain type of aero-engine, and significantly improving the service life of the transmission system.

[0028] Example 2 This embodiment addresses the issue of stress concentration and weak resistance to error disturbances in traditional straight-line splines used in heavy-duty aerospace transmission systems. It employs a five-node cubic B-spline modification technique to optimize the tooth surface structure, resolving problems such as stress concentration and weak resistance to error disturbances inherent in traditional straight-line modification techniques. This approach adapts to the stringent operating conditions of heavy-duty aerospace transmissions. The specific implementation process is as follows: I. Target Audience and Basic Parameters This embodiment is adapted for splines used in heavy-duty aircraft transmission systems. These splines undertake heavy-duty, high-frequency transmission tasks, requiring extremely high resistance to tooth surface disturbances, contact strength, and transmission efficiency. The core basic parameters are as follows: tooth width L = 60mm, core contact area width of the tooth surface is 10mm, and the left end trimming amount is set. =0.03mm, right end trimming amount =0.015mm, tangent point position parameter =25mm, left end center shaping parameter =0.015mm, right end center shaping parameter =0.013mm, ensuring a smooth transition of the shaping curve and coordinated parameter adaptation. Specific implementation steps

[0029] Step 1: Determine the reshaping area Based on the five-node cubic B-spline theory, the spline tooth surface modification interval is defined, and the stretching coordinates are set along the tooth width direction. The range of the reshaping interval is clearly defined as follows: ,in Corresponding to the left end of the spline, Corresponding to the right end of the spline, it precisely covers the 10mm wide core contact area of ​​the tooth surface and the entire tooth width range, providing a basic framework for subsequent parameter verification, curve modeling and machining planning; Step 2: Set key control parameters Set key control parameters for the shaping curve, including the shaping amount at the left end. Left end middle shaping parameters Tangent point position parameters Right end middle section shaping parameters Right end shaping amount Based on the operating requirements of heavy-duty aerospace transmission splines, the core parameters were determined as follows: =0.03mm, =0.015mm, =25mm, =0.015mm, =0.013mm, all parameters are within the coordinated constraint range, which is suitable for heavy-duty transmission strength requirements; Step 3: Establish a parameter coordination constraint mechanism and solve for the parameters. based on , , Given three parameters, establish a parameter coordination constraint mechanism and verify it. , The reasonableness of the selected value is determined by the following process: Determine the degree of the B-spline curve The number of secondary control points is 5. Based on the open interval node vector rules, the node vector is set. It is suitable for the segment uniformity requirement of 60mm tooth width; First, the cubic B-spline basis function values ​​are calculated using the Cox-deBoor recursive form. The zero-degree basis function is calculated using the formula... Higher-order basis functions are defined by recursive formulas. Calculations, obtained synchronously: The corresponding basis function values ​​of each parameter ( , respectively corresponding ); Tangent point position basis functions Calculations yielded , , , ; Secondly, calculate the derivative coefficients of the basis functions. and the linear combination coefficient term combining the shaping amounts at both ends Substitute , , , , , By simplifying the formula and substituting the basis function values ​​at each discrete measuring point, the following results are obtained for the full tooth width at 10 discrete measuring points (0, 6, 12, 18, 24, 25, 30, 40, 50, 60 mm). and ,For example: hour, ,because The value is 0, which does not meet the constraint formula premise, so it is directly excluded. Seven valid measurement points are calculated. , ; Subsequently, through the constraint formula Calculated Reasonable range is ,verify =0.015mm falls within this range, so substitute it into the formula. Correct the original setting deviation and confirm. The optimal value is 0.049mm, and all shaping parameters have been calibrated. Step 4: Generate the final shaping curve equation The calibrated shaping parameters and corresponding cubic B-spline basis functions are used. Substituting into the general expression for B-spline curves, we obtain the final modified curve equation. This equation can accurately output the position of any position in the tooth width direction. shaping amount This provides a clear mathematical basis for tooth surface forming.

[0030] Step 5: Tooth surface forming, machining adaptation, and accuracy verification Based on the final profile curve equation, the complete profiled tooth surface is formed by projecting from the tooth tip to the tooth root. Simultaneously, machining scheme planning, error adaptation, and accuracy testing are carried out. The specific process is as follows: Modeling and simulation verification: All calibrated shaping parameters were imported into UG 3D modeling software to generate a five-node cubic B-spline shaping spline model; the model was imported into Hypermesh software for mesh generation preprocessing, and then Abaqus finite element software was used to simulate heavy-load conditions to verify the spline's resistance to disturbances caused by 0.01° angular misalignment and ±0.008mm tooth thickness error. Based on the simulation results, the machining parameters were optimized and the optimal cutting scheme was determined. CNC machining: A special shaping tool is used in conjunction with a CNC milling machine to control the cutting speed at 35m / min and the feed rate at 0.04mm / r. The tool moves along the preset B-spline curve trajectory to accurately machine the shaped tooth surface. Parametric design further enhances the adaptability to tooth thickness and pitch angle errors, ensuring transmission stability under heavy load conditions. Precision testing: After machining, 10 measuring points (including core contact area points) were evenly selected in the z-direction of the tooth width using a coordinate measuring machine. The test results showed that the tooth direction error was ≤0.005mm and the surface roughness was ≤0.8μm, which met the preset precision requirements. At the same time, the maximum contact stress on the tooth surface was measured to be 286.3MPa by a pressure sensor, which met the strength requirements and showed no interference.

[0031] III. Application Effects When applied to heavy-duty aerospace transmission systems, this modified spline assembly exhibits excellent adaptability to various operating conditions: it can stably withstand 0.01° angular misalignment and ±0.008mm tooth thickness error, and the maximum contact stress is reduced by 64.8% compared to traditional straight modified splines, effectively avoiding tooth surface fatigue damage under heavy-duty conditions; during operation, the transmission is smooth and interference-free, and the transmission efficiency is improved by 8% compared to traditional structures, fully meeting the stringent requirements of heavy-duty aerospace transmissions and significantly enhancing the reliability and service life of the transmission system.

[0032] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A spline topology optimization method based on B-spline curves, characterized in that, Includes the following steps: Step 1: Determine the spline tooth surface modification interval based on the five-node cubic B-spline theory; Step 2: Set the key control parameters for the shaping curve, including the left end shaping amount. Left end middle shaping parameters Tangent point position parameters Right end middle shaping parameters Right end shaping amount ; Step 3: Establish a parameter coordination constraint mechanism and utilize... , , Lock left-side center shaping parameters The numerical range is then used to derive the shaping parameters for the middle right end through formulas. The value; Step 4: Based on all the determined shaping parameters and interpolation basis functions, substitute them into the general expression for B-spline curves to obtain the final shaping curve equation; Step 5: Based on the final modified curve equation, project and form the complete modified tooth surface, plan the B-spline trajectory machining scheme and set the cutting parameters, adapt the machining and assembly errors through parametric design, and combine simulation optimization and detection to ensure that the tooth surface accuracy meets the standards.

2. The spline topology optimization method based on B-spline curves according to claim 1, characterized in that, In step one, the shaping interval is set along the tooth width direction, and the tooth width direction is defined as the stretching coordinate. The reshaping range is ,in, It represents the width of the spline teeth.

3. The spline topology optimization method based on B-spline curves according to claim 2, characterized in that, In step two, represent The amount of reshaping at the location, represent The amount of reshaping at the location, The coordinates of the point of tangency between the profile curve and the tip line of the spline tooth in the tooth width direction are represented, with a value range of [value missing]. .

4. The spline topology optimization method based on B-spline curves according to claim 3, characterized in that, In step three, the shaping parameters of the left middle section are locked. The formula for the numerical range is expressed as follows: ； in, represent The lower limit of the value; represent The upper limit of the possible values; This means that the constraint conditions apply to all discrete measurement points in the tooth width direction; Represents the range of reshaping intervals The first A discrete sampling point; Representing the The coefficient terms derived from the basis functions of each discrete sampling point; Representing the A linear combination coefficient term of discrete sampling points combined with the shaping amounts at both ends.

5. The spline topology optimization method based on B-spline curves according to claim 4, characterized in that, The coefficient terms derived from the basis functions The calculation formula is: ； in, Represents corresponding The cubic B-spline basis function values; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; represent At the tangent point The cubic B-spline basis function value at the location.

6. The spline topology optimization method based on B-spline curves according to claim 4, characterized in that, The linear combination coefficient term that combines the shaping amounts at both ends The calculation formula is: ； in, Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at point; Represents corresponding The cubic B-spline basis function values; represent At the tangent point The cubic B-spline basis function value at the location.

7. The spline topology optimization method based on B-spline curves according to claim 5, characterized in that, The formula for calculating the basis function value of the cubic B-spline is as follows: ； ； ； in, represent Number B-spline basis functions; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; Representative Table Number B-spline basis function values; Represents the first node in the node vector Each node value; Represents the first node in the node vector Each node value; represent Number The basis function value of the B-spline and Together they constitute the recursive computation terms of higher-order basis functions.

8. The spline topology optimization method based on B-spline curves according to claim 7, characterized in that, The B-spline basis functions are given in Cox-deBoor recursive form, where the zero-order basis function is defined as follows: ； in, The parameter variables represent the B-spline curve.

9. The spline topology optimization method based on B-spline curves according to claim 7, characterized in that, In step three, the shaping parameters of the right-end middle section are derived. The numerical derivation formula is as follows: 。 10. A spline topology optimization method based on B-spline curves according to claim 9, characterized in that, In step four, the final formula expression for the modified curve equation is: ； in, Represents the position in the tooth width direction The corresponding shaping amount value; Representing the The shaping parameters corresponding to each control point.

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