Complete tooth groove second-order continuous parameterization modeling and hybrid multi-solution inverse method based on radius

Abstract

The application discloses a complete tooth groove second-order continuous parameterization modeling and a mixed multi-solution reverse solving method based on a radius, and belongs to the technical field of gear digitization modeling. The method sequentially splices a left involute, a dedendum section and a right involute into a single global parameter interval to form a unified segmented parameter context. A quintic Hermite base function is used to construct a dedendum transition curve, so that the dedendum transition curve matches the positions, first-order derivatives and second-order derivatives of adjacent involutes at both ends. A central radius correction term and a longitudinal downward exploration term with zero to second-order quantities of zero are superimposed to realize decoupling of shape adjustment in the middle of the dedendum and endpoint constraint. In response to a target radius, an analytical inverse solution is performed on the involute section, and a full-interval or local multi-interval bisection numerical inverse solution is performed on the dedendum section according to the monotonicity of the radius sequence. After mapping the local solutions of each section to the global parameter interval, the solutions are sorted, joint tolerance is removed and output. The application realizes unified parameterization of forward evaluation and reverse solving of a complete tooth groove, and improves modeling accuracy and reverse solving stability.

Description

Technical Field

[0001] This invention belongs to the field of gear digital modeling and design technology, specifically involving a complete tooth groove second-order continuous parametric modeling and a radius-based hybrid multi-solution inverse method, which can be applied to engineering processes such as tooth profile discretization, machining area division, toolpath calculation, interference checking, three-dimensional mesh generation, and detection section intersection in gear CAM software. Background Technology

[0002] Involute cylindrical gears are the most widely used transmission components in the field of mechanical transmission. Their tooth groove cross-section consists of three parts: the left involute tooth profile, the tooth root transition section, and the right involute tooth profile. In the digital design and manufacturing process of gears, accurate geometric modeling of the complete tooth groove is the foundation for subsequent tooth surface discretization, finite element analysis, CNC machining toolpath planning, and tooth profile accuracy inspection. For a long time, how to achieve accurate, efficient, and consistent modeling of the entire tooth groove area has been one of the core issues in the field of gear digital technology.

[0003] In existing gear tooth groove modeling techniques, the left and right involute sections are usually directly generated using involute analytical equations, resulting in high modeling accuracy. However, accurate modeling of the tooth root transition curve is relatively difficult. Existing techniques mainly construct tooth root transition curves in the following ways: First, using a simplified circular arc to replace the tooth root transition curve. While this method is simple to implement, it cannot accurately reflect the shape of the actual machined tooth profile, introducing significant errors in tooth root bending stress analysis. Second, establishing parametric equations for the tooth root transition curve based on the generating machining principle can obtain a relatively accurate tooth root shape, but usually requires specific derivation for different tool parameters and machining conditions. Third, using the Hermite interpolation method to construct the tooth root transition surface, obtaining a smooth transition by interpolating discrete points in the tooth root region.

[0004] However, the aforementioned existing technologies still have the following shortcomings.

[0005] (1) Each segment of the tooth profile maintains an independent parameter system, resulting in poor interface consistency. In the existing technology, the left involute, the root transition section, and the right involute are usually described by different parameterization methods. The involute part uses its development angle or pressure angle as a parameter, while the root transition section uses normalized arc length or interpolation parameters as variables. Each segment maintains its own local parameters and local boundary definitions, resulting in different segment judgment logics for different interfaces such as point value calculation, derivative solution, discrete sampling, and radius inverse calculation. The attribution method or parameter direction of the same joint in different interfaces is often inconsistent, which requires the upper-level application (such as mesh generation and toolpath planning) to frequently perform parameter conversion when switching between different areas of the tooth groove, increasing the complexity of the system and the risk of errors.

[0006] (2) Only point values ​​or tangential continuity are guaranteed; second-order parameters change abruptly at the joint. In existing technologies for constructing tooth root transition curves using Hermite interpolation, some schemes employ cubic or quartic Hermite interpolation. Cubic Hermite interpolation can only control the endpoint positions and the first derivative, and can achieve C 1 While continuous (tangentially continuous) interpolation is possible, the second derivative cannot be controlled. Fourth-order Hermite interpolation, although offering more degrees of freedom, differs fundamentally in its basis function form and the number of endpoint constraints compared to fifth-order Hermite. When only point-value continuity or tangential continuity is used as the joint constraint, the second-order parameters (curvature-related quantities) of the tooth root segment and the left and right involutes at the joint cannot achieve the same caliber matching. In subsequent tooth surface normal estimation, discrete sampling density calculation, curvature analysis, and numerical machining trajectory planning, abrupt changes in the second-order differential components directly affect the stability and accuracy of the calculation results.

[0007] (3) Shape adjustment inevitably disrupts established joint constraints. In existing tooth root shape adjustment methods, when it is necessary to change the tooth root center radius or the downward shape to meet specific tooth root strength or meshing performance requirements, the coordinates of the tooth root curve control points or the boundary conditions are usually directly modified. This adjustment method inevitably changes the position, first derivative, or second derivative of the joint, thereby disrupting the established joint continuity constraints. To re-satisfy the joint conditions, the entire tooth root segment needs to be reconstructed and iteratively solved, resulting in low design efficiency.

[0008] (4) The inverse calculation method for a given radius lacks a targeted segmented processing strategy. In gear engineering applications, it is often necessary to inversely calculate the corresponding parameter positions on the tooth groove based on a given target radius (e.g., root circle radius, measurement section radius, or grid partition boundary radius). Existing technologies typically employ two methods to handle such inverse calculation problems: first, using only the analytical expression of a single-sided involute for inverse solution, completely ignoring possible intersections on the root transition section; second, using a uniform numerical iteration method to solve the entire tooth groove. The former will miss valid solutions when the target radius falls within the root section range; the latter does not consider the special mathematical properties of the root section radius function, i.e., the change of the root radius along the parameter may be non-monotonic. When the root section radius function is non-monotonic, a single iteration may miss solutions, converge to an incorrect branch, or generate multiple nearly repeated invalid solutions.

[0009] (5) There is a lack of a unified deduplication mechanism for the solution results of different segments.

[0010] When the left and right involutes and tooth root segments are solved independently, each segment near the splice boundary may return numerically very similar solutions. These solutions actually correspond to the same geometric position on the tooth groove, but are output repeatedly because they belong to different segments. Existing technologies lack a unified processing mechanism for global filtering, sorting, and joint tolerance deduplication of multi-segment solution results. This leads to unstable and redundant boundary parameters being received by the upper-level mesh generation, toolpath partitioning, or tooth profile detection process, affecting the reliability of subsequent processing.

[0011] In summary, existing gear tooth groove modeling techniques often employ different models and parameter systems for forward geometric evaluation (point values, derivatives, discrete sampling) and reverse radius calculation. There is a lack of a unified modeling framework that can describe the complete tooth groove while simultaneously supporting high-precision forward evaluation and stable reverse calculation. This invention addresses these technical problems. Summary of the Invention

[0012] In view of this, the purpose of this invention is to provide a complete tooth groove second-order continuous parametric modeling and a radius-based hybrid multi-solution inverse method, aiming to establish a unified piecewise parametric complete tooth groove model, realize the matching of the position, first and second derivatives of the tooth root segment and the involute at the joint with the same diameter, and on this basis, provide a radius hybrid inverse method that combines involute analysis and tooth root numerical analysis, so as to balance the accuracy of forward calculation and the stability of inverse solution.

[0013] To achieve the above objectives, the present invention provides the following technical solution: A complete second-order continuous parametric modeling of tooth grooves and a radius-based hybrid multi-solution inverse method include the following steps: Step S1: Obtain the gear geometric principal quantities, which include at least the addendum circle radius, the root reference radius, the left involute base circle radius, and the right involute base circle radius. Perform a validity check on non-finite values, non-positive values, or missing values. The left and right base circle radii are allowed to be unequal to support asymmetrical tooth profiles. When the root reference radius is a missing or non-positive value, construct a minimum center radius based on the smaller base circle radius and a preset radial retraction amount. Step S2: Calculate the left involute span and the right involute span according to the tooth tip circle radius and the base circle radius of each side respectively. Determine the tooth root transition segment span according to the tooth root radial retraction and the involute spans on both sides. Then, sequentially splice the left involute segment, the tooth root transition segment and the right involute segment into a single global parameter interval of the total span to form a unified segmented parameter context. Step S3: Establish local parameter mapping for each segment in the context of the unified segmentation parameters: use reverse local parameter mapping for the left involute segment to make the local parameters of the left involute near the tooth root joint approach zero; use normalized parameter mapping for the tooth root segment; use positive local parameter mapping for the right involute segment to make the local parameters of the right involute near the tooth root joint approach zero. Step S4: Construct a tooth root transition curve using a fifth-order Hermite basis function, so that the tooth root transition curve matches the position of the left involute endpoint, the first derivative and the second derivative with respect to the global parameters at the left end, and matches the position of the right involute endpoint, the first derivative and the second derivative with respect to the global parameters at the right end, forming a complete tooth groove model with continuous second-order parameters. Step S5: Superimpose a center radius correction term and a longitudinal downward extension term on the base curve of the tooth root transition curve. The function values, first derivatives, and second derivatives of the center radius correction term and the longitudinal downward extension term at both ends of the tooth root segment are all zero, thereby adjusting the shape of the middle part of the tooth root without changing the established position and matching relationship of the first derivative and the second derivative at both ends. Step S6: In response to the input target radius, perform analytical inverse solution on the left involute segment and the right involute segment respectively. The analytical inverse solution calculates local parameters based on the base circle radius of each side and the target radius, and receives the local candidate solution of the involute on that side only when the calculated local parameters are within the effective span of that side. Step S7: Perform numerical inverse solution on the tooth root transition segment, including: generating multiple sampling points within the normalized parameter interval of the tooth root segment, calculating the radius value at each sampling point, and determining the monotonicity of the radius sequence; when the radius sequence is approximately monotonic, if the endpoint residuals are out of sign, perform a bisection search on the entire tooth root interval; when the radius sequence is not monotonic, sequentially identify the sampling near-zero points and the local intervals with adjacent residuals that are out of sign, and perform a bisection search on each local interval with out-of-sign to obtain local candidate solutions for the tooth root segment; Step S8: Map each segment of local candidate solutions to the global parameter range, including mapping the left involute local solution, the right involute local solution, and the tooth root segment normalization solution to global parameter values ​​according to a preset correspondence, and merging the mappings of each segment to form a candidate solution set; Step S9: Perform effective range filtering on the candidate solution set, retain only finite values ​​located within the global parameter range, sort by global parameter values, and eliminate adjacent duplicate solutions with a preset global parameter tolerance, and output the deduplicated global candidate solution set.

[0014] Furthermore, in step S2, the span of the left involute... and the span of the involute on the right They are represented as follows:

[0015] in: The radius of the tooth tip circle; The radius of the left base circle; The radius of the right base circle; The total span satisfies:

[0016] in: This refers to the span of the tooth root transition section.

[0017] Furthermore, in step S3, the local parameters of each segment are mapped as follows: On the left At that time, the local parameters of the involute on the left side ; In the root segment At that time, the normalized parameters of the tooth root segment ; On the right section At that time, the local parameters of the involute on the right side ; in: To unify the parameters for the entire tooth groove; The span of the involute on the left side; This refers to the span of the tooth root transition section.

[0018] Furthermore, in step S4, the tooth root transition curve is constructed in the following manner: Normalize the tooth root segment to an interval It uses six fifth-order Hermite basis functions:

[0019]

[0020]

[0021] Constructing the basic horizontal curve:

[0022] in: , and These are Hermite basis functions for the left-hand position, left-hand first order, and left-hand second order, respectively; , and These are Hermite basis functions for the right-hand side, the first-order left-hand side, and the second-order left-hand side, respectively. , These are the x-coordinates of the left and right connection points of the tooth root, respectively; , Converted to The endpoint transverse second derivative under the parameters; the basic transverse curve at both ends and The position and second derivative of the adjacent involute endpoints are matched; Normalized parameters for the tooth root segment; The x-coordinates of the left and right connection points are respectively taken as: , The transverse second derivatives of the endpoints are respectively taken as... , The lateral radius coordinates of the target midpoint of the tooth root The tooth root reference radius Take as valid ,exist When a value is missing, it is recalculated by shrinking back to the smaller base circle radius, where .

[0023] Furthermore, the coordinates of the points on the tooth root curve are:

[0024] in: This is the center radius correction function; It is a vertical downward function; This is a correction factor; This represents the longitudinal downward amplitude of the tooth root; and:

[0025]

[0026] in: The coordinates of the lateral radius of the target point at the midpoint of the tooth root; Normalized parameters for the tooth root segment; Center radius correction function and vertical downward function Satisfying and At this point, the function value, first derivative, and second derivative are all zero, that is:

[0027]

[0028] Thus the center radius correction term and vertical downward exploration item The superposition does not change the basic transverse curve. The endpoint conditions have been established at both ends; at the left seam. and right seam At this point, the root segment and the adjacent involute satisfy the following:

[0029] in: For the tooth root curve; for Lateral involute; To unify the parameters for the entire tooth groove; Corresponding to the left seam, Corresponding to the right seam.

[0030] Furthermore, in step S6, the coordinates of the points on the left and right involutes are calculated using the following unified formula:

[0031] in: for The radius of the base circle on the side; ,for Dimensionless expansion parameters of the lateral involute; For lateral symbols, take the left side. Take the right side ; for Local parameters of the lateral involute; Corresponding to the left seam, Corresponding to the right seam; the radius of the involute point satisfies... This relationship forms the basis for the analytical inverse solution; The first derivatives of the left and right involutes with respect to global parameters under unified parameters are calculated using the chain rule, introducing a negative direction on the left or a positive direction on the right; the second derivatives are directly reused from the analytical second derivatives because there is a linear mapping between local and global parameters.

[0032] Furthermore, in step S6, the analytical inverse solution calculates the local parameters according to the following formula:

[0033] in: , Corresponding to the left seam, Corresponding to the right seam; The target radius is given only if the area under the square root is non-negative and At that time, local candidate solutions for the involute on that side are received.

[0034] Furthermore, in step S7, the method for performing numerical inverse kinematics on the tooth root transition segment is as follows: Take within the normalized parameter range of the tooth root segment Each sub-interval forms a sampling parameter sequence. ,in Calculate the radius value at each sampling point. ; Define tooth root radius residual ; When all adjacent differences All are not less than When determining the radius sequence, it is either monotonically non-decreasing or all of them are no greater than 1. When determining the radius sequence, it is determined that it is monotonically non-increasing, where: To pre-determine the monotonicity tolerance; When the radius sequence is determined to be approximately monotonic, check the endpoint residuals. and Are the signs different? If they are different, then apply to the entire interval. Perform a binary search; When the radius sequence is determined to be non-monotonic, each sampling point is checked sequentially: when When the solution is directly recorded as a near-zero candidate solution, then: The preset radius residual tolerance; when Time interval Perform binary search on each of the local root regions; Each iteration of the binary search takes:

[0035] And based on the sign of the midpoint residual, retain the half of the interval that still contains the root, that is, if Then let Otherwise until Or it may reach the preset iteration limit; in: , and These are the lower bound, upper bound, and midpoint of the bisection interval, respectively.

[0036] Furthermore, in step S8, the method for mapping each segment of local candidate solutions to the global parameter range is as follows: Local solution of the involute on the left Mapped to ; Local solution of the involute on the right side Mapped to ; Tooth root segment unification and resolution Mapped to .

[0037] Furthermore, in step S9: The method for eliminating adjacent duplicate solutions with a preset global parameter tolerance is as follows: the sorted adjacent global candidate solutions satisfy... When, eliminate the last candidate solution, where: For the first There are 10 candidate solutions; Deduplication tolerance for preset global parameters; Output the global candidate solution set according to the preset maximum number of output solutions. Output the sorted first A maximum number of output solutions; in one implementation, the maximum number of output solutions. Output at most two global candidate solutions in descending order; Effective range filtering method: when the target radius satisfy The solution set is returned directly when the time is empty, where: The radius of the tooth tip circle; To predetermine the outer circle comparison tolerance; for negative numbers, zero values, and non-finite target radius values, an empty solution set is also returned; when the tooth root span When the value is less than the preset positive lower limit, it is restricted to the positive lower limit to avoid the denominator being zero when differentiating the normalized mapping and the chain rule.

[0038] Furthermore, in non-inverse calculation scenarios, the unified piecewise parameter context established in steps S1 to S5 is directly used to calculate the point values ​​of the complete tooth groove and the first and second derivatives with respect to the global parameters. This ensures that the radius function used in the inverse calculation process and the point value function used in geometric sampling come from the same completely consistent piecewise geometric context. The point values ​​of the tooth root segment and their first and second derivatives with respect to the global parameters are uniformly obtained from the same tooth root curve expression according to the chain rule.

[0039] in: For the tooth root curve; This refers to the span of the tooth root transition section; To unify the parameters for the entire tooth groove; Normalized parameters for the tooth root segment; Avoid using point-value formulas and derivative formulas from different models.

[0040] The beneficial effects of this invention are as follows: This invention presents a complete second-order continuous parametric modeling method for tooth grooves and a radius-based hybrid multi-solution inverse method. Through nine steps, it constructs a complete tooth groove parametric framework that shares the same piecewise context from forward modeling to inverse solution, achieving the following technical effects: (1) Unify parameter semantics and eliminate interface ambiguity: The left involute (reverse mapping), tooth root segment (normalized mapping), and right involute (forward mapping) are sequentially spliced ​​into a single global parameter range. Point values, derivatives and inverse calculations share the same context, avoiding confusion in the joint attribution caused by inconsistent local parameter directions in the segmented tooth profile. (2) The second-order parameter is continuous, which improves the numerical stability: the fifth Hermite basis function matches the position, first derivative and second derivative at both ends of the tooth root, so that the second-order differential component has no abrupt change at the splice, which is beneficial to the subsequent normal estimation, discrete sampling density control and stable calculation of machining trajectory. (3) Decoupling of shape control and endpoint constraints: The function values, first derivative and second derivative of the center radius correction term and the longitudinal downward extension term are all zero at both ends, so that when adjusting the downward extension depth or center radius of the tooth root, the established joint continuity constraint is not destroyed and the endpoint conditions do not need to be solved again. (4) Hybrid inverse solution improves solution stability: the involute segment adopts analytical inverse solution, which does not require iteration; the tooth root segment is divided according to the monotonicity of the radius sequence: when it is monotonic, the whole interval is divided into two, and when it is non-monotonic, multiple local intervals with different signs are identified one by one and divided into two; this avoids missing solutions or converging to the wrong branch under the non-monotonic radius function in a single iteration. (5) Output specifications, adapted to upper-level engineering: After global mapping, range filtering, sorting and joint tolerance deduplication of the three local solutions, stable boundary parameters are output, which can be directly used for mesh partitioning, toolpath interval division or detection of cross-section intersections, reducing post-processing conversion errors. Attached Figure Description

[0041] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 The flowchart shows the complete second-order continuous parametric modeling of tooth grooves and the radius-based hybrid multi-solution inverse method of the present invention. Detailed Implementation

[0042] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0043] like Figure 1 As shown in the figure, the complete second-order continuous parametric modeling of tooth groove and the method for inverse calculation of multiple solutions based on radius in this embodiment are characterized by the following steps.

[0044] Step S1: Obtain the gear geometric principal quantities. The geometric principal quantities include at least the addendum circle radius, the root reference radius, the left involute base circle radius, and the right involute base circle radius, and perform a validity check on non-finite values, non-positive values, or missing values; the left base circle radius and the right base circle radius are allowed to be unequal to support asymmetrical tooth profiles; when the root reference radius is a missing value or a non-positive value, a minimum center radius is constructed based on the smaller base circle radius and a preset radial retraction amount.

[0045] Specifically, in one feasible approach, the geometric principal quantities of the gear cross-section are read from a gear design parameter database or a user input interface, and these geometric principal quantities include at least the addendum circle radius. Reference radius of tooth root Radius of the left involute base circle and the radius of the involute base circle on the right The radii of the left and right base circles can be equal or unequal; when they are unequal, it is called an asymmetrical tooth profile.

[0046] After obtaining the aforementioned geometric principal variables, a validity check is performed on each radius value. The validity check includes: checking whether each value is a finite number (excluding infinity and non-numeric values ​​like NaN), whether it is a positive number (excluding zero and negative values), and whether there are any missing values. If a geometric principal variable is non-finite, non-positive, or missing, it is processed according to a preset strategy. Specifically: When the radius of the left base circle Or the radius of the base circle on the right side If an error occurs, a valid involute tooth profile cannot be constructed. In this case, an error flag can be returned or subsequent calculations can be terminated. When the tooth root reference radius When the value is missing or non-positive, it is not used directly; instead, it is determined based on the smaller base circle radius. A minimum center radius is constructed using the preset radial retraction amount. The minimum center radius is constructed as follows: [The text abruptly ends here, so the translation stops.] Subtract a preset positive shrinkage amount (e.g., 0.1 to 0.3 times the module) so that the tooth root curve still has a calculable geometric scale.

[0047] In one implementation, the tooth root reference radius When effective, the target lateral radius coordinates of the midpoint of the tooth root By limiting the tooth root reference radius to no more than the smaller base circle radius, the geometric rationality of the tooth root curve is ensured.

[0048] Step S2: Calculate the span of each segment and form a unified segmentation parameter context. Calculate the left involute span and the right involute span based on the tooth tip circle radius and the base circle radii of each side, respectively. Determine the tooth root transition segment span based on the tooth root radial retraction and the involute spans on both sides. Sequentially concatenate the left involute segment, the tooth root transition segment, and the right involute segment into a single global parameter interval for the total span, forming a unified segmentation parameter context.

[0049] Based on the addendum circle radius and left and right base circle radii obtained in step S1, calculate the left involute span using the following formula. and the span of the involute on the right :

[0050] in: The radius of the tooth tip circle; The radius of the left base circle; The radius of the right base circle is given.

[0051] When the radius of the addendum circle is less than or equal to the radius of the base circle on one side, the square root of the corresponding involute span is a non-positive number. A span of 0 indicates that there is no effective involute segment on that side.

[0052] Tooth root transition span The determination method is as follows: First, based on the tooth root reference radius... With smaller base circle radius The difference is used to calculate the radial retraction at the tooth root, i.e. (when Effective and (when) or preset retraction amount (when) (When missing). Then, the tooth root span is determined based on the radial retraction amount and the involute span on both sides. In one embodiment, Take the sum of the radial retraction and approximately 0.25 times the sum of the left and right spans, that is:

[0053] in: This is a preset positive lower limit value used to ensure that the tooth root span is not too small. The purpose of setting this positive lower limit is to avoid problems with the denominator in subsequent steps such as normalization mapping and chain rule differentiation. It is zero.

[0054] The left involute segment, the tooth root transition segment, and the right involute segment are sequentially spliced ​​together to form the total span. Single global parameter range This forms a unified segmentation parameter context. This context records the span and segmentation boundaries of each segment. , , , ) and the corresponding segment type identifiers (left involute segment, tooth root segment, right involute segment).

[0055] Step S3: Establish local parameter mapping for each segment within the unified segmented parameter context. Use reverse local parameter mapping on the left involute segment to make the local parameters of the left involute near the tooth root joint approach zero; use normalized parameter mapping on the tooth root segment; use forward local parameter mapping on the right involute segment to make the local parameters of the right involute near the tooth root joint approach zero.

[0056] Specifically, for the uniform input parameters First, restrict it to the global parameter range. Inside. Then according to The segmented interval is mapped using the corresponding local parameters: (1) In the left segment At that time, it belongs to the left involute segment. Using reverse local parameter mapping, the local parameters of the left involute are:

[0057] This reverse mapping makes when (That is, when the left involute is close to the tooth root joint) ;when (That is, when the left involute is close to the tooth tip joint) This directional arrangement causes the local parameters of the left involute near the tooth root joint to approach zero, which is semantically consistent with the local parameters of the right involute near the tooth root joint.

[0058] (2) At the root segment At this point, it belongs to the tooth root transition section. Using normalized parameter mapping, the normalized parameters for the tooth root section are:

[0059] This mapping will remove the tooth root segment from the global parameters. Mapping to normalized interval This makes the left and right ends of the tooth root segment correspond to respectively and This facilitates subsequent construction using standardized Hermite base functions.

[0060] (3) In the right segment At that time, it belongs to the right involute segment. Using positive local parameter mapping, the local parameters of the right involute are:

[0061] This positive mapping makes when (That is, when the right involute is close to the tooth root joint) ;when (That is, when the right involute is close to the tooth tip joint) The local parameters of the involute on the right side, near the tooth root joint, also approach zero.

[0062] in: To unify the parameters for the entire tooth groove; The span of the involute on the left side; This refers to the span of the tooth root transition section.

[0063] The above three mapping relationships together constitute the local parameter transformation rules of the unified piecewise parameter context, so that forward evaluation and reverse solution share the same parameter semantics.

[0064] Step S4: Construct a tooth root transition curve using a fifth-order Hermite basis function, such that the tooth root transition curve matches the position of the left involute endpoint, the first derivative and the second derivative with respect to global parameters at the left end, and matches the position of the right involute endpoint, the first derivative and the second derivative with respect to global parameters at the right end, thus forming a complete tooth groove model with continuous second-order parameters.

[0065] First, normalize the tooth root segment to an interval. At the left end of the tooth root segment At this point, the position of the endpoint of the left involute needs to be matched. First derivative with respect to global parameters and second derivative ; at the right end of the tooth root segment At this point, the position of the endpoint of the right involute needs to be matched. First derivative with respect to global parameters and second derivative .

[0066] Since the first derivative of the involute at the base circle connection end (i.e., the end closest to the tooth root) degenerates to zero (with respect to the development angle or local parameter), and the first derivative at the tooth root endpoint under the global parameter is zero (because the derivative of the local parameter with respect to the global parameter is a constant, and remains zero after multiplication), the first derivative conditions at both ends simplify to zero. However, the second derivative conditions need to be retained and matched under the global parameter caliber to ensure that the second parameter quantities are of the same caliber at the splice.

[0067] In this embodiment, the tooth root transition curve is constructed as follows: Using six fifth-order Hermite basis functions:

[0068]

[0069]

[0070] These basis functions have the following properties: and Control the positions of both ends, and Controlling the first derivative at both ends, and Control the second derivative at both ends.

[0071] Since the first derivatives at both ends are zero, the basis functions related to the first derivative are... and The coefficient is zero. Construct the basic transverse curve:

[0072] in: , , which are the abscissas of the left and right connection points of the tooth root (the ordinates are determined by the ordinates of the involute at the endpoint of the base circle; due to the orientation arrangement, the ordinates at this point can be calculated using the involute formula). , Converted to Endpoint transverse second derivative under parameters; Since the first derivative is zero, it does not appear in the expression for the basic transverse curve. and item.

[0073] For the ordinate of the tooth root curve, fifth-order Hermite interpolation is also used. The ordinates at both ends are determined by the ordinates of the involute endpoints. The first derivative at both ends is zero, and the second derivative at both ends matches the ordinate component of the second derivative of the involute endpoints. Let the basic ordinate curve be denoted as... Then the tooth root basic curve is .

[0074] This construction method ensures that the left seam... and right seam At this point, the position of the tooth root segment and the adjacent involute, as well as the first and second derivatives with respect to the global parameters, are matched to form a complete tooth groove model with continuous second-order parameters.

[0075] Step S5: Superimpose a center radius correction term and a longitudinal downward extension term on the base curve of the tooth root transition curve. The function values, first derivatives, and second derivatives of the center radius correction term and the longitudinal downward extension term at both ends of the tooth root segment are all zero, thereby adjusting the shape of the middle part of the tooth root without changing the established position and matching relationship of the first derivative and the second derivative at both ends.

[0076] To adjust the shape of the tooth root midpoint without changing the conditions at both endpoints established in step S4, two correction terms are superimposed on the base curve: (1) Center radius correction term: superimposed along the horizontal (x direction) ,in:

[0077] This function is in and The function value, first derivative, and second derivative are all zero. The correction factor is the target lateral radius at the midpoint of the tooth root. Sure:

[0078] That is, by adjusting Make the lateral coordinate of the midpoint of the tooth root reach the target value. . Reference radius at the tooth root Take when valid ,exist When a value is missing, it is recalculated by shrinking back to the smaller base circle radius, where .

[0079] (2) Vertical downward term: superimposed along the longitudinal direction (y direction) ,in:

[0080] This function is also in and The function value, first derivative, and second derivative are all zero. The downward amplitude is a non-negative number, which can be determined by the weighted sum of the radial retraction and the floor level.

[0081] The coordinates of the final point of the tooth root curve are:

[0082] in: This is the center radius correction function; It is a vertical downward function; This is a correction factor; This represents the longitudinal downward thrust amplitude of the tooth root; This is a correction term for the center radius; Vertical downward exploration item.

[0083] Due to the center radius correction function and vertical downward function The function value, first derivative, and second derivative are all zero at both ends:

[0084]

[0085] Therefore, the function values, first derivatives, and second derivatives of the center radius correction term and the longitudinal downward extension term at both ends are all zero. Their superposition will not change the matching relationship of the two ends, first derivative, and second derivative established in step S4. Designers can independently adjust these parameters according to the requirements of tooth root strength and meshing performance. or This allows you to change the shape of the middle part of the tooth root without having to resolve the endpoint conditions.

[0086] At the left seam and right seam At this point, the root segment and the adjacent involute satisfy the following:

[0087] in: For the tooth root curve; for Lateral involute; To unify the parameters for the entire tooth groove; Corresponding to the left seam, Corresponding to the right seam.

[0088] The first and second derivatives of the tooth root segment with respect to global parameters are obtained uniformly from the same tooth root curve expression using the chain rule:

[0089]

[0090] in: For the tooth root curve; This refers to the span of the tooth root transition section; To unify the parameters for the entire tooth groove; These are normalized parameters for the tooth root segment. This avoids the point-value formula and derivative formula originating from different models.

[0091] Step S6: In response to the input target radius, perform analytical inverse solutions on the left and right involute segments respectively. The analytical inverse solutions calculate local parameters based on the base circle radius of each side and the target radius, and receive the local candidate solution of the involute on that side only when the calculated local parameters are within the effective span of that side.

[0092] In gear engineering applications, when a target radius is given When this is done, it is necessary to reverse-engineer all the tooth grooves that satisfy the condition that the radius is equal to... The parameter positions. For involute segments, due to the distinct geometric properties of the involute: the square of the distance from any point on the involute to the center of the gear is equal to the sum of the square of the base circle radius and the square of the unfolded length, i.e. Therefore, it can be solved directly through analysis.

[0093] For a given target radius The local parameters of the left and right involute curves are calculated using the following formula:

[0094]

[0095] For each side, the local solution is accepted only if the following two conditions are met: (1) The radical sign is not negative, that is Otherwise, there is no point with the target radius on the involute line on that side; (2) The local parameters are located within the effective span on this side, i.e. Otherwise, the point is located outside the tooth tip circle or outside the effective range of the involute.

[0096] When the target radius is smaller than the base circle radius on a certain side, there is no solution for the involute on that side, but there may still be a solution for the tooth root segment. Therefore, the process continues to execute instead of terminating entirely.

[0097] In one implementation, the coordinates of the points on the left and right involutes are calculated using the following unified formula:

[0098] in: for The radius of the base circle on the side; ,for Dimensionless expansion parameters of the lateral involute; For lateral symbols, take the left side. Take the right side ; for Local parameters of the lateral involute; Corresponding to the left seam, Corresponding to the right seam; the radius of the involute point satisfies... This relationship forms the basis for the analytical inverse solution.

[0099] The first derivative of the global parameters is calculated using the chain rule, since the left side uses a reverse mapping ( The right side uses a positive mapping () The chain rule naturally introduces the negative direction on the left and the positive direction on the right. The second derivative directly reuses the analytic second derivative because there is a linear mapping between local and global parameters.

[0100] Step S7: Perform numerical inverse solution on the tooth root transition segment, including: generating multiple sampling points within the normalized parameter interval of the tooth root segment, calculating the radius value at each sampling point, and determining the monotonicity of the radius sequence; when the radius sequence is approximately monotonic, if the endpoint residuals are out of sign, perform a bisection search on the entire tooth root interval; when the radius sequence is not monotonic, sequentially identify the sampling near-zero point and the local intervals with adjacent residuals that are out of sign, and perform a bisection search on each local interval with out-of-sign to obtain local candidate solutions for the tooth root segment.

[0101] Unlike involute segments, tooth root segments do not have a simple analytical inverse solution formula and require numerical methods. However, the variation of the tooth root radius function with respect to parameters may be non-monotonic, thus necessitating targeted numerical strategies.

[0102] The specific implementation steps are as follows: First, in the normalized parameter range of the tooth root segment Internal extraction Each sub-interval forms a sequence of sampling parameters:

[0103] In one implementation method That is, 97 sampling points. The number of sampling points can be chosen to balance accuracy requirements and computational efficiency. More sampling points make it easier to find local intervals with different signs, but the amount of computation will also increase accordingly.

[0104] Calculate the radius value at each sampling point:

[0105] And define the tooth root radius residual:

[0106] Then determine the monotonicity of the radius sequence. When all adjacent differences All are not less than The radius sequence is determined to be monotonically non-decreasing when all adjacent differences are... All are not greater than The radius sequence is determined to be monotonically non-increasing. Where: To preset the monotonicity judgment tolerance (in one implementation, take...) ), used to suppress pseudo-nonmonotonicity caused by floating-point numerical noise.

[0107] Different solution strategies are executed based on the monotonicity determination result: (1) When the determination radius sequence is approximately monotonic, check the endpoint residuals. and Are the signs different? If they are different, it means that in... If a root exists within the interval, perform a binary search on the entire interval. In each iteration of the binary search, the midpoint is taken:

[0108] And retain the half of the interval that still contains the root based on the sign of the midpoint residual. If Then let Otherwise Iterate until... Or it may reach a preset iteration limit. In one implementation, the iteration limit for bisection of the entire interval in the monotonic case is 60 times. Wherein: This is the preset radius residual tolerance; , and These are the lower bound, upper bound, and midpoint of the bisection interval, respectively.

[0109] (2) When the radius sequence is determined to be non-monotonic, check each sampling point sequentially. When it is, it is directly recorded as a near-zero candidate solution. When, indicate the interval If the residuals at both ends are different in sign and there is a transverse root, this interval is treated as a local root interval and a bisection search is performed on each interval. The upper limit of the bisection search iteration for each local interval with different signs can be less than the bisection of the entire interval, which is 40 times in one implementation.

[0110] In the above solution process, For the preset radius residual tolerance (in one embodiment, take...) This is used to determine whether the convergence accuracy is met.

[0111] This diversion strategy can effectively handle the case where the tooth root radius function is non-monotonic: non-monotonic means that the tooth root radius first increases and then decreases or first decreases and then increases along the parameter direction. At this time, the tooth root segment and the target radius may have two or more intersection points. A single full interval bisection is prone to missing solutions or converging to the wrong branch, while by bisectioning multiple local intervals with opposite signs, all cross-type intersection points can be obtained.

[0112] Step S8: Map each segment of local candidate solutions to the global parameter range, including mapping the left involute local solution, the right involute local solution, and the tooth root segment normalization solution to global parameter values ​​according to a preset correspondence, and merging the mappings of each segment to form a candidate solution set.

[0113] The local solutions for the left and right involutes obtained in step S6 and the normalized solution for the tooth root segment obtained in step S7 reside in different local parameter spaces and need to be uniformly mapped to global parameters. Only on the axis can uniform sorting and deduplication be performed.

[0114] The mapping method for each segment's local candidate solution is as follows: (1) Local solution of the involute on the left side Mapped to global parameters:

[0115] This mapping is the left segment local mapping in step S3. The inverse function of .

[0116] (2) Local solution of the involute on the right side Mapped to global parameters:

[0117] This mapping is the right segment local mapping in step S3. The inverse function of .

[0118] (3) Unification of tooth root segment Mapped to global parameters:

[0119] This mapping is the local mapping of the tooth root segment in step S3. The inverse function of .

[0120] All global parameter values ​​obtained from the above three mappings are merged to form a candidate solution set. At this point, each element in the candidate solution set is located within the global parameter range. The scalar values ​​within have a unified dimension and semantics, which facilitates subsequent unified processing.

[0121] Step S9: Perform effective range filtering on the candidate solution set, retain only finite values ​​located within the global parameter range, sort by global parameter values, and eliminate adjacent duplicate solutions with a preset global parameter tolerance, and output the deduplicated global candidate solution set.

[0122] First, the candidate solution set is filtered for an effective range, retaining only candidate solutions that meet the following conditions: Located within the global parameter range, i.e. ; For finite values, exclude infinity or non-numeric values ​​(NaN).

[0123] In one implementation, it also includes fast no-solution determination: when the target radius satisfy The solution set is returned directly when the time is empty. To preset the outer circle comparison tolerance (in one embodiment, take...) This means that if the target radius significantly exceeds the tip circle radius, no solution is required. For negative and zero target radii, an empty solution set is also returned.

[0124] Then, the filtered candidate solutions are sorted in ascending order of global parameter values ​​to obtain the sorted sequence. .

[0125] Next, joint tolerance deduplication is performed. Because the left and right involute sections and the tooth root segment may each yield numerically very similar solutions independently at the splicing boundary (for example, the left involute section at...). The returned solution and the root segment are at The solutions returned at each location actually correspond to the same geometric position, and these nearly duplicated solutions need to be removed. The deduplication rule is: for adjacent global candidate solutions after sorting... and If the following conditions are met:

[0126] Then eliminate the second candidate solution. (reserve ).in To determine the deduplication tolerance for preset global parameters (in one implementation, take...) ).

[0127] Finally, output the maximum number of solutions as preset. Output the sorted first One solution. In one implementation, to ensure compatibility with existing upper-layer interfaces (e.g., mesh partitioning interfaces requiring dual solution output), take... Output at most two global candidate solutions in descending order (i.e., take the last two after sorting). In other implementations, different settings can be configured based on the capacity of the upper-layer interface. value.

[0128] In addition, when the tooth root span When the value is less than the preset positive lower limit, it is restricted to that positive lower limit to avoid the denominator being zero when differentiating the normalization mapping in step S3 and the chain rule in step S5, thus ensuring the numerical stability of the calculation process.

[0129] When applied to an embodiment of a non-inverse calculation scenario, based on the above steps, in a non-inverse calculation scenario (i.e., when the radius inverse calculation of steps S6 to S9 is not required), the unified segmented parameter context established in steps S1 to S5 can be directly used to calculate the point value of the complete tooth groove and the first and second derivatives of the global parameters.

[0130] Specifically, for any input global parameter First, the local parameter mapping in step S3 is used to determine the segment to which it belongs and calculate the corresponding local parameters. Then, the corresponding segment evaluator is called: If it is located on the left involute segment, call the involute formula. and its derivative; If it is located at the tooth root segment, call the tooth root curve formula. and its derivative; If it is located on the right side of the involute segment, call the involute formula. and its derivative.

[0131] The first and second derivatives of the tooth root segment with respect to global parameters are obtained uniformly from the same tooth root curve expression according to the chain rule of the following formula:

[0132]

[0133] The radius function used in the reverse process (steps S6 to S9) and the point value function used in the forward geometric sampling come from the same piecewise geometric context, thus avoiding the data inconsistency problem that occurs when the point value formula and the derivative formula come from different models.

[0134] When applied to the implementation of boundary and exception handling, the following boundary and exception handling should also be performed during the implementation of the above steps: (1) When the input target radius For non-finite values, non-positive values, or values ​​greater than the tip circle radius Add outer circle comparison tolerance If the condition is met, return an empty solution set directly.

[0135] (2) When the target radius is smaller than the radius of the base circle on a certain side, the analytical solution of the involute on that side does not exist (the square root is negative), and the analytical solution of the involute on that side is not generated, but the search for the intersection point on the tooth root segment continues.

[0136] (3) When the left and right base circle radii or the tooth tip circle radii are insufficient to form an effective involute span (i.e. Time span The corresponding span is set to zero, and the unified context determines whether a valid tooth root segment can still be constructed.

[0137] (4) When the tooth root span If the value is too small (less than the preset positive lower limit), limit it to the positive lower limit to avoid the denominator being zero when differentiating the normalization mapping in step S3 and the chain rule in step S5.

[0138] (5) When the residual exactly meets the tolerance at the sampling point ( The sampled solution is recorded directly without the need for additional binary search.

[0139] (6) When the solution is located near the junction of the involute and the tooth root, and is simultaneously identified by two segments (e.g., the left involute is at the junction of the involute and the tooth root), The solution at the root segment is in The solution at the given location is processed through the joint tolerance deduplication mechanism in step S9, retaining only one valid solution.

[0140] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A complete second-order continuous parametric modeling method for tooth grooves and a radius-based hybrid multi-solution inverse method, characterized in that: Includes the following steps: Step S1: Obtain the gear geometric principal quantities, which include at least the addendum circle radius, the root reference radius, the left involute base circle radius, and the right involute base circle radius. Perform a validity check on non-finite values, non-positive values, or missing values. The left and right base circle radii are allowed to be unequal to support asymmetrical tooth profiles. When the root reference radius is a missing or non-positive value, construct a minimum center radius based on the smaller base circle radius and a preset radial retraction amount. Step S2: Calculate the left involute span and the right involute span according to the tooth tip circle radius and the base circle radius of each side respectively. Determine the tooth root transition segment span according to the tooth root radial retraction and the involute spans on both sides. Then, sequentially splice the left involute segment, the tooth root transition segment and the right involute segment into a single global parameter interval of the total span to form a unified segmented parameter context. Step S3: Establish local parameter mapping for each segment in the context of the unified segmentation parameters: use reverse local parameter mapping for the left involute segment to make the local parameters of the left involute near the tooth root joint approach zero; use normalized parameter mapping for the tooth root segment; use positive local parameter mapping for the right involute segment to make the local parameters of the right involute near the tooth root joint approach zero. Step S4: Construct a tooth root transition curve using a fifth-order Hermite basis function, so that the tooth root transition curve matches the position of the left involute endpoint, the first derivative and the second derivative with respect to the global parameters at the left end, and matches the position of the right involute endpoint, the first derivative and the second derivative with respect to the global parameters at the right end, forming a complete tooth groove model with continuous second-order parameters. Step S5: Superimpose a center radius correction term and a longitudinal downward extension term on the base curve of the tooth root transition curve. The function values, first derivatives, and second derivatives of the center radius correction term and the longitudinal downward extension term at both ends of the tooth root segment are all zero, thereby adjusting the shape of the middle part of the tooth root without changing the established position and matching relationship of the first derivative and the second derivative at both ends. Step S6: In response to the input target radius, perform analytical inverse solution on the left involute segment and the right involute segment respectively. The analytical inverse solution calculates local parameters based on the base circle radius of each side and the target radius, and receives the local candidate solution of the involute on that side only when the calculated local parameters are within the effective span of that side. Step S7: Perform numerical inverse solution on the tooth root transition segment, including: generating multiple sampling points within the normalized parameter interval of the tooth root segment, calculating the radius value at each sampling point, and determining the monotonicity of the radius sequence; when the radius sequence is approximately monotonic, if the endpoint residuals are out of sign, perform a bisection search on the entire tooth root interval; when the radius sequence is not monotonic, sequentially identify the sampling near-zero points and the local intervals with adjacent residuals that are out of sign, and perform a bisection search on each local interval with out-of-sign to obtain local candidate solutions for the tooth root segment; Step S8: Map each segment of local candidate solutions to the global parameter range, including mapping the left involute local solution, the right involute local solution, and the tooth root segment normalization solution to global parameter values ​​according to a preset correspondence, and merging the mappings of each segment to form a candidate solution set; Step S9: Perform effective range filtering on the candidate solution set, retain only finite values ​​located within the global parameter range, sort by global parameter values, and eliminate adjacent duplicate solutions with a preset global parameter tolerance, and output the deduplicated global candidate solution set.

2. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S2, the span of the left involute and the span of the involute on the right They are represented as follows: in: The radius of the tooth tip circle; The radius of the left base circle; The radius of the right base circle; The total span satisfies: in: This refers to the span of the tooth root transition section.

3. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S3, the local parameters of each segment are mapped as follows: On the left At that time, the local parameters of the involute on the left side ; In the root segment At that time, the normalized parameters of the tooth root segment ; On the right section At that time, the local parameters of the involute on the right side ; in: To unify the parameters for the entire tooth groove; The span of the involute on the left side; This refers to the span of the tooth root transition section.

4. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S4, the tooth root transition curve is constructed in the following manner: Normalize the tooth root segment to an interval It uses six fifth-order Hermite basis functions: Constructing the basic horizontal curve: in: , and These are Hermite basis functions for the left-hand position, the first-order left-hand position, and the second-order left-hand position, respectively. , and These are Hermite basis functions for the right-hand side, the first-order left-hand side, and the second-order left-hand side, respectively. , These are the x-coordinates of the left and right connection points of the tooth root, respectively; , Converted to The endpoint transverse second derivative under the parameters; the basic transverse curve at both ends and The position and second derivative of the adjacent involute endpoints are matched; Normalized parameters for the tooth root segment; The x-coordinates of the left and right connection points are respectively taken as: , The transverse second derivatives of the endpoints are respectively taken as... , The lateral radius coordinates of the target midpoint of the tooth root The tooth root reference radius Take as valid ,exist When a value is missing, it is recalculated by shrinking back to the smaller base circle radius, where .

5. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S5, the coordinates of the points on the tooth root curve are: in: This is the center radius correction function; It is a vertical downward function; This is a correction factor; This represents the longitudinal downward amplitude of the tooth root; and: in: The coordinates of the lateral radius of the target point at the midpoint of the tooth root; Normalized parameters for the tooth root segment; Center radius correction function and vertical downward function Satisfying and At this point, the function value, first derivative, and second derivative are all zero, that is: Thus the center radius correction term and vertical downward exploration item The superposition does not change the basic transverse curve. The endpoint conditions have been established at both ends; at the left seam. and right seam At this point, the root segment and the adjacent involute satisfy the following: in: For the tooth root curve; for Lateral involute; To unify the parameters for the entire tooth groove; Corresponding to the left seam, Corresponding to the right seam.

6. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S6, the coordinates of the points on the left and right involutes are calculated using the following unified formula: in: for The radius of the base circle on the side; ,for Dimensionless expansion parameters of the lateral involute; For lateral symbols, take the left side. Take the right side ; for Local parameters of the lateral involute; Corresponding to the left seam, Corresponding to the right seam; the radius of the involute point satisfies... This relationship forms the basis for the analytical inverse solution; The first derivatives of the left and right involutes with respect to global parameters under unified parameters are calculated using the chain rule, introducing a negative direction on the left or a positive direction on the right; the second derivatives are directly reused from the analytical second derivatives because there is a linear mapping between local and global parameters.

7. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multiple solution inverse method according to claim 1, characterized in that: In step S6, the analytical inverse solution calculates the local parameters according to the following formula: in: , Corresponding to the left seam, Corresponding to the right seam; The target radius is given only if the area under the square root is non-negative and At that time, local candidate solutions for the involute on that side are received.

8. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S7, the method for numerical inverse kinematics of the tooth root transition segment is as follows: Take within the normalized parameter range of the tooth root segment Each sub-interval forms a sampling parameter sequence. ,in Calculate the radius value at each sampling point. ; Define tooth root radius residual ; When all adjacent differences All are not less than When determining the radius sequence, it is either monotonically non-decreasing or all of them are no greater than 1. The radius sequence is determined to be monotonically non-increasing when: To pre-determine the monotonicity tolerance; When the radius sequence is determined to be approximately monotonic, check the endpoint residuals. and Are the signs different? If they are different, then apply to the entire interval. Perform a binary search; When the radius sequence is determined to be non-monotonic, each sampling point is checked sequentially: when When the solution is directly recorded as a near-zero candidate solution, then: For the preset radius residual tolerance; when Time interval Perform binary search on each of the local root regions; Each iteration of the binary search takes: And based on the sign of the midpoint residual, retain the half of the interval that still contains the root, that is, if Then let Otherwise until Or it may reach the preset iteration limit; in: , and These are the lower bound, upper bound, and midpoint of the bisection interval, respectively.

9. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multiple solution inverse method according to claim 1, characterized in that: In step S8, the method for mapping each segment of local candidate solutions to the global parameter range is as follows: Local solution of the involute on the left Mapped to ; Local solution of the involute on the right side Mapped to ; Tooth root segment unification and resolution Mapped to .

10. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In step S9: The method for eliminating adjacent duplicate solutions with a preset global parameter tolerance is as follows: the sorted adjacent global candidate solutions satisfy... When, eliminate the last candidate solution, where: For the first There are 10 candidate solutions; Deduplication tolerance for preset global parameters; Output the global candidate solution set according to the preset maximum number of output solutions. Output the sorted first A maximum number of output solutions; in one implementation, the maximum number of output solutions. Output at most two global candidate solutions in descending order; Effective range filtering method: when the target radius satisfy The solution set is returned directly when the time is empty, where: The radius of the tooth tip circle; To predetermine the outer circle comparison tolerance; for negative numbers, zero values, and non-finite target radius values, an empty solution set is also returned; when the tooth root span When the value is less than the preset positive lower limit, it is restricted to the positive lower limit to avoid the denominator being zero when differentiating the normalized mapping and the chain rule.

11. The complete tooth groove second-order continuous parametric modeling and radius-based hybrid multi-solution inverse method according to claim 1, characterized in that: In non-inverse calculation scenarios, the unified piecewise parameter context established in steps S1 to S5 is directly used to calculate the point values ​​of the complete tooth groove and the first and second derivatives with respect to global parameters. This ensures that the radius function used in the inverse calculation process and the point value function used in geometric sampling come from the same completely consistent piecewise geometric context. The point values ​​of the tooth root segment and their first and second derivatives with respect to global parameters are uniformly obtained from the same tooth root curve expression according to the chain rule. in: For the tooth root curve; This refers to the span of the tooth root transition section; To unify the parameters for the entire tooth groove; Normalized parameters for the tooth root segment; Avoid using point-value formulas and derivative formulas from different models.

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