Ship non-developable thickened gear tooth surface point cloud modeling method

Abstract

The present application relates to the technical field of shipbuilding, in particular to a ship non-involute variable-thickness gear tooth surface point cloud modeling method, comprising: establishing the conjugate tooth surface equation of the involute variable-thickness gear and the non-involute variable-thickness gear, and constructing a third-order nonlinear equation set; solving the unknown variable data corresponding to the non-involute variable-thickness gear addendum circle; solving the unknown variable data corresponding to different radii of the non-involute variable-thickness gear middle section tooth profile; based on the Lagrange interpolation method, constructing the iteration initial value of each section unknown variable of the full tooth surface according to the obtained data set; solving the non-involute variable-thickness gear tooth surface point cloud, and taking the involute development angle range of the non-involute variable-thickness gear as the numerical solution judgment condition in the solving process to exclude non-target solutions, compared with the prior art, the present application can improve the problems of instability and complex process of the existing ship variable-thickness gear, has high stability, fast convergence speed and can meet the high-precision processing demand.

Description

Technical Field

[0001] This invention relates to the field of shipbuilding technology, specifically to a point cloud modeling method for the tooth surface of non-involute variable thickness gears for ships that can improve the instability and complex process of existing marine variable thickness gears, has high stability, fast convergence speed, and can meet the requirements of high-precision machining. Background Technology

[0002] In marine gearboxes, gears operate under prolonged low-speed, heavy-load conditions characterized by high corrosion and frequent impacts, making them susceptible to typical failures such as tooth surface fatigue pitting and tooth bending fatigue fracture. This severely challenges their meshing contact performance. Non-involute variable-thickness gears can mesh with involute variable-thickness gears to achieve line contact transmission, with both having conjugate tooth surfaces. Non-involute variable-thickness gear pairs overcome the limitations of point contact in involute variable-thickness gear pairs in transmission, exhibiting superior contact performance. The tooth surface equation of a non-involute variable-thickness gear can be obtained from the tooth surface equation of an involute variable-thickness gear through coordinate transformation. Figure 1 For the spatial transformation coordinate system of intersecting axis non-involute variable thickness gears; Figure 2 This is the spatial transformation coordinate system for the staggered-axis non-involute variable-thickness gear. The non-involute variable-thickness gear pair consists of one involute variable-thickness gear and one non-involute variable-thickness gear, whose tooth surfaces are conjugate surfaces. The tooth surface equation of the involute variable-thickness gear is easy to obtain, while the tooth surface equation of the non-involute variable-thickness gear needs to be obtained by spatial coordinate transformation using the tooth surface equation of the involute variable-thickness gear. Figure 1 and Figure 2 These are the spatial transformation coordinate systems for the conjugate meshing of intersecting axis non-involute variable thickness gear pairs and staggered axis non-involute variable thickness gear pairs.

[0003] Figure 1 In the coordinate system This is a fixed coordinate system for the involute gear with varying thickness, which moves together with the gear. Let be the initial coordinate system of the involute gear with varying thickness, and let be the natural coordinate system. This coordinate system is related to the initial position of the involute gear with varying thickness and remains fixed in space. The coordinate system is a fixed coordinate system for a non-involute variable-thickness gear, and it moves with the gear; coordinate system The initial coordinate system for the non-involute variable thickness gear is the natural coordinate system; the axial intersection angle of the intersecting shaft variable thickness gear pair is... The involute thickens as it wraps around the gear. Rotation Non-involute thickened gears surrounding Rotation The coordinate transformation process for an intersecting shaft non-involute variable thickness gear pair is as follows: . Figure 2 For the spatial transformation coordinate system of the staggered-axis non-involute variable-thickness gear, the coordinate system is... The coordinate system is a fixed coordinate system for the involute gear with varying thickness, and it moves with the gear; coordinate system The initial coordinate system for the involute gear with varying thickness is the natural coordinate system; coordinate system The coordinate system is a fixed coordinate system for a non-involute variable-thickness gear, and it moves with the gear; coordinate system The initial coordinate system for the non-involute variable thickness gear is the natural coordinate system; the crossing angle of the staggered shaft variable thickness gear pair is... , and The inclination angle of the tooth line on the pitch plane when each tooth meshes with the common rack; and These are the rotational speeds of the two gears around their respective axes, and the rotational speeds of the involute-thickened gears around their axes. Rotation Non-involute thickened gears surrounding Rotation The coordinate transformation process for a staggered-axis non-involute variable-thickness gear pair is also as follows: .

[0004] There are two existing methods for obtaining point clouds of the tooth surface of non-involute variable-thickness gears: direct and indirect methods. The direct method uses numerical methods to directly solve for the point cloud of the tooth surface of non-involute variable-thickness gears from the tooth surface equation. However, since the tooth surface equation is a nonlinear transcendental equation system, the existence of multiple solutions often leads to the obtained numerical solution being non-target. Furthermore, unreasonable initial values ​​during iteration can cause convergence failure. The indirect method starts with the involute variable-thickness gear, comparing the tooth surface morphology differences between the involute variable-thickness gear and the non-involute variable-thickness gear to obtain the tooth profile difference and tooth direction difference. Then, using the tooth profile difference and tooth direction difference, a shaping approach is adopted to obtain the point cloud of the non-involute variable-thickness gear tooth surface. This process is complex, computationally cumbersome, and has accumulated errors. Specifically, the tooth surface equation of an intersecting-axis non-involute variable-thickness gear is: (1), In the formula It is the base circle radius of the involute gear with increasing thickness. It is the axial angle of an intersecting non-involute variable-thickness gear pair. The base circle pitch of the involute gear with increasing thickness. It is the development angle of the involute gear with increasing thickness. It is the rotation angle of the involute gear with increasing thickness. It is the starting angle of the involute gear with increasing thickness. The starting angle of a non-involute variable thickness gear. It is the base cylinder helix angle of the involute gear with increasing thickness. It is the transmission ratio of the gear pair.

[0005] The tooth surface equation of a non-involute variable-thickness gear with staggered shafts is: (2), In the formula, represents the crossover angle of the non-involute variable thickness gear pair with cross-axis. It is the shortest distance between the two intersecting axes of the thickened gear pair.

[0006] The tooth surface is divided into sections along the tooth width direction (Z-axis direction). The tooth height direction is divided into portions. The total number of tooth surface dot clouds is [number]. There are two traditional methods for obtaining point clouds of the tooth surface of non-involute variable thickness gears: direct method and indirect method. (a) Direct method: Based on the derived tooth surface equation of the non-involute variable thickness gear, numerical methods such as the quasi-Newton method are used to solve the nonlinear equation system. The general process is as follows: Figure 3 As shown, the superscript... This represents different cross-sections of the gear, and its value range is: subscript This represents the radius of the circle containing each point on the gear tooth profile, and its value range is... Using the tooth surface equations of the non-involute variable thickness gear shown in equations (1) and (2), an iterative equation system is constructed, and the solution of the full tooth surface point cloud can be completed through nested loops. (b) Indirect method: The indirect method requires obtaining the tooth surface point cloud of the involute variable thickness gear based on the tooth surface equation of the involute variable thickness gear, and then calculating the difference in tooth shape and tooth direction between the involute and non-involute variable thickness gears by comparing the tooth surface morphology. Finally, based on the idea of ​​shape modification, the existing tooth surface point cloud of the involute variable thickness gear is processed to obtain the tooth surface point cloud of the non-involute variable thickness gear, such as Figure 4 As shown. Summary of the Invention

[0007] This invention addresses the shortcomings and deficiencies of existing technologies by proposing a point cloud modeling method for the tooth surface of marine non-involute variable thickness gears, which is highly stable, has a fast convergence speed, and can meet the requirements of high-precision machining.

[0008] This invention achieves its purpose through the following measures: A method for modeling point clouds of the tooth surface of a marine non-involute variable-thickness gear, characterized by the following steps: Step 1: Establish the conjugate tooth surface equations for involute variable thickness gears and non-involute variable thickness gears, and construct a third-order nonlinear equation system for solving the point cloud of the tooth surface of non-involute variable thickness gears. Step 2: Solve for the unknown variable data corresponding to the addendum circle of the non-involute variable thickness gear; Step 3: Solve for the unknown variable data corresponding to different radii of the tooth profile at the intermediate section of the non-involute variable thickness gear; Step 4: Based on the Lagrange interpolation method, construct the iterative initial values ​​of the unknown variables of each section of the full tooth surface according to the datasets obtained in Step 2 and Step 3; Step 5: Based on the obtained initial values ​​of the iteration, use a three-level nested loop to solve the point cloud of the tooth surface of the non-involute variable thickness gear. In the process of solving, the range of the involute development angle of the non-involute variable thickness gear is used as the judgment condition for the numerical solution to exclude non-target solutions.

[0009] In step 2 of this invention, it is recorded For involute gears with varying thicknesses, the addendum circle radii are given by different cross-sections, and the corresponding involute development angles are given by different cross-sections. From equation (6), we get: (6), in the formula This refers to the pressure angle corresponding to the addendum circle of different cross-sections. The involute development angle cannot be less than 0. According to the gear meshing principle, the involute development angle corresponding to the addendum circle of a non-involute variable thickness gear is necessarily smaller than the involute development angle corresponding to the involute variable thickness gear. That is, the involute development angle of a non-involute variable thickness gear... Satisfy the relationship shown in equation (7): (7) The tangent of the pressure angle at the middle of the tooth profile of the involute variable thickness gear is used as the initial iterative value of the development angle corresponding to the tip circle of different sections of the non-involute variable thickness gear, denoted as Let the rotation angle corresponding to the middle of the tooth width of the involute gear be . , as the initial value for the iteration of the rotation angle corresponding to the addendum circle of different cross sections of the non-involute variable thickness gear; the initial value for the iteration of the starting angle is set to ,remember The addendum circle radii are for different cross-sections of non-involute variable thickness gears. Given the axial position values ​​of different sections of a non-involute variable thickness gear, then the unknown variable corresponding to the addendum circle of the non-involute variable thickness gear is... , , The solution can be obtained as follows: The solution process includes two nested loops. The inner loop determines whether the numerical solution obtained by the quasi-Newton method is the target solution. If it is, it is output directly; otherwise, it is sorted by size. The step size changes the initial value of the starting angle, and the solution is repeated. The outer loop is used to traverse and solve for the unknown variables of different cross sections. Each time the outer loop is completed, the position of the cross section needs to be changed. The value and the corresponding tooth tip circle radius The initial value of the position variable in each subsequent loop is the target solution obtained in the previous outer loop, except for the first outer loop.

[0010] The specific solution process in step 2 of this invention is as follows: Step 2-1: Initialize the outer loop variable. Let the outer loop variable be i, which is used to represent different cross sections of the gear. Set the initial value to 0 and the value range is 0 ≤ i ≤ M, where M is the total number of cross sections. After solving each cross section, execute i = i + 1 until all cross sections are traversed. Step 2-2: Set the initial values ​​for the iteration of the unknown variables. For the current section i, set the initial values ​​for the iteration of the three unknown variables: the initial value of the involute development angle. ,in The initial value of the involute development angle corresponding to the current tooth tip circle; initial value of the starting angle. , The initial value of the starting angle corresponding to the middle of the tooth width; the initial value of the rotation angle. Initially set to ; Step 2-3: Initialize the inner loop variable k. Let the inner loop variable be k, which is used to adjust the initial value of the starting angle. The initial value is set to k = 0, where k is an integer and 0 ≤ k ≤ n, and n is the preset maximum number of adjustments. At the same time, reset the current initial value of the starting angle. According to the formula Perform initial settings; Steps 2-4: Solve the nonlinear equation system using the quasi-Newton method, with the currently given initial values. The following third-order nonlinear equations are solved using the quasi-Newton method: Position Equation 1: , The radius of the tooth tip circle at the current cross-section. Position Equation 2: , This represents the axial position of the current cross-section. Conjugate meshing equation: Solve for the unknown variable value corresponding to the current numerical solution. .

[0011] Step 2-5: Determine if the numerical solution is the target solution, and check the involute development angle in the numerical solution obtained in Step 2-4. Does the objective solution meet the criteria? ,in Let be the development angle of the tip circle corresponding to the current section of the involute thickened gear. If the above conditions are met, the numerical solution is the target solution, and we jump to step 2-7. If the above conditions are not met, the numerical solution is a non-target solution, and we execute step 2-6. Steps 2-6: Adjust the initial value of the starting angle and solve again: Let the inner loop variable k = k + 1, and then... Update the initial angle value, that is, change it by a fixed step size of 2π / n. Then return to steps 2-4 and use the quasi-Newton method again with the new initial angle value; Step 2-7: Save the target solution for the current cross-section: Save the current numerical solution as the target solution corresponding to the tooth tip circle of the cross-section. ; Step 2-8: Update the initial value of the position variable for the next iteration: To speed up convergence, the solution obtained from the current section is used as the initial value of the position variable for the next section, i.e., let: ; Step 2-9: Determine if all cross sections have been traversed. Let the outer loop variable i = i + 1, and check if i satisfies i ≤ M. If it does, it means there are still cross sections that have not been solved. Return to step 2-2 and solve the unknown variables of the tooth tip circle of the next cross section. If it does not satisfy the condition, it means that the unknown variables of the tooth tip circle of all cross sections have been solved. Proceed to the next step. Step 2-10: The solution for the unknown variable data corresponding to the tooth tip circle is complete, and the process ends.

[0012] In step 3 of this invention, based on the tooth surface equations of the involute variable thickness gear and the non-involute variable thickness gear, the tooth surfaces of the involute variable thickness gear and the non-involute variable thickness gear are conjugate tooth surfaces. The tooth surface equation of the non-involute variable thickness gear is determined by the variables of the involute variable thickness gear itself. , ,and To illustrate, according to the gear meshing principle, when the tooth tip of a non-involute thickened gear engages, the meshing point is close to the base circle of the involute thickened gear. This should be the minimum value within the range of variation of the development angle of a non-involute variable thickness gear, denoted as the development angle corresponding to the base circle of the involute variable thickness gear. Since the tooth profile of the involute thickened gear is an involute and the base circle is the starting circle of the involute, the development angle corresponding to the base circle is 0. Let the development angle corresponding to the base circle of the non-involute variable thickness gear be . Since the involute thickened gear and the non-involute thickened gear have similar tooth profile heights, the relationship shown in equation (8) exists: (8) Due to The value is always 0, therefore the development angle of the involute gear corresponding to the base circle of the non-involute thickening gear is: (9) In order to include the actual range of the development angle of non-involute variable thickness gears, the hypothetical maximum value of the development angle corresponding to different cross-sectional profiles of non-involute variable thickness gears is defined as follows: ,but Recommended: (10), where C is the amplification factor, which is taken as 1.1-1.2; For non-involute thickened gears, the range of the development angle corresponding to the tooth profile is within the range shown in equation (11): (11), Let the radius of the circle containing each point on the tooth profile at the midpoint of the tooth width of the non-involute variable-thickness gear be . Let the axial position of the middle part of the tooth width of the non-involute variable-thickness gear be denoted as . If the initial values ​​of the unknown variables are set in the same way as those set in the first step of calculating the addendum circle, then the unknown variables corresponding to different radii of the tooth profile at the middle of the tooth width of the non-involute variable-thickness gear are... , , The solution can be obtained, and the solution process differs from that in step 2 in that only the radius needs to be changed after each completion of the outer loop. The value is sufficient to trigger the next outer loop, where... for The value corresponding to the cross section at the middle of the tooth width.

[0013] Step 4 of this invention specifically involves: based on the dataset of unknown variables corresponding to the addendum circle of the non-involute variable-thickness gear and the dataset of unknown variables corresponding to the tooth profile of the intermediate section obtained above, firstly, for different radii of the tooth profile of the intermediate section... Lagrangian fitting is performed on the corresponding dataset of unknown variables; then, based on the tooth tip circle of each section... Based on the corresponding dataset of unknown variables and referring to the Lagrange equation for the intermediate section, the Lagrange equations for other cross sections are given. The specific process is as follows: First, the Lagrange equation is obtained by fitting the dataset of unknown variables corresponding to the tooth profile of the intermediate section of the non-involute variable-thickness gear: (12) Based on equation (12), the initial value equations for the other sections can be obtained as follows: (13) In the formula It is the involute development angle of the involute thickening gear corresponding to the tooth tip circle of the middle section; It is the rotation angle of the involute gear with increasing thickness corresponding to the tooth tip circle of the middle section; It is the starting angle of the involute thickening gear corresponding to the tip circle of the intermediate cross section.

[0014] Step 5 of this invention specifically involves: using a three-layer nested loop to solve the point cloud of the tooth surface of a non-involute variable-thickness gear. The inner loop is used to determine the rationality of the numerical solution, with equation (11) as the determination condition. If the numerical solution is determined to be non-target solution, the initial value of the starting angle will be changed according to the step size, and the numerical solution will be performed again. If the numerical solution is determined to be target solution, the inner loop will be exited to continue solving the next point. The middle loop is used to solve the point cloud of the tooth profile, specifically to solve the point cloud of the tooth profile at different cross sections. Each time a first loop is completed... The corresponding section on the th cross-section After solving for each tooth profile point, it needs to be changed once. The value is performed for the first time The first section Solving for each tooth profile point; the outer loop is used to change the cross section, when the first tooth profile point is completed. After solving for all tooth profile points on a cross section, the axial position parameters need to be changed. The value of the first Solving for all tooth profile points in each cross section; after each inner loop is completed, that is, after each tooth profile point is solved numerically, the 3D coordinate data of the tooth profile point is exported.

[0015] The specific steps of step 5 of this invention are as follows: Step 5-1: Initialize the loop variables. Let the outer loop variable be i, which represents different cross sections of the gear. Its value range is 0 ≤ i ≤ M, where M is the total number of cross sections. Let the middle loop variable be j, which represents each point on the tooth profile on the same cross section. Its value range is 0 ≤ j ≤ N, where N is the total number of tooth profile points on a single cross section. Let the inner loop variable be k, which represents the number of iterations. The initial value is set to k=0, where k is an integer and 0 ≤ k ≤ n. Step 5-2: Given the initial values ​​of the unknown variables, calculate and assign the initial values ​​of the unknown variables based on the pre-constructed Lagrange interpolation function, using the current section index i and tooth profile point index j: Initial value of the involute development angle. Initial value of starting angle, initial value of rotation angle ; Step 5-3: Solve the nonlinear equation system using the quasi-Newton method. With a given initial value, solve the third-order nonlinear iterative equation system of the point cloud of the non-involute variable-thickness gear tooth surface using the quasi-Newton method. The specific equation system is as follows: Position Equation 1: ; Position Equation 2: ; Conjugate meshing equation: ; Solve for the unknown variable value corresponding to the current numerical solution. ; Step 5-4: Determine if the numerical solution is the target solution, and determine the involute development angle in the obtained numerical solution. Does the objective solution meet the criteria? ,in The maximum value of the development angle of the non-involute variable-thickness gear in the current section is assumed. If the above conditions are met, the numerical solution is the target solution, and the process jumps to step 5-6; if the above conditions are not met, the numerical solution is a non-target solution, and step 5-5 is executed. Step 5-5: Adjust the initial value of the starting angle and solve again: Let the inner loop variable k = k + 1, change the initial value of the starting angle according to the set step size, and then return to step 5-3 to solve the nonlinear equation system again using the quasi-Newton method with the new initial value of the starting angle. Steps 5-6: Output the target solution and 3D coordinates of the current tooth profile point: Save the current numerical solution as the target solution for this tooth profile point and record the target solution: Simultaneously calculate and output the three-dimensional coordinate data of that point: X coordinate: , Y coordinate: , Z-coordinate: ; Step 5-7: Determine whether the tooth profile points of the current section have been traversed. Let the tooth profile point index j = j + 1, and determine whether j satisfies j ≤ N: If it satisfies, it means that there are still tooth profile points in the current section that have not been solved, and return to step 5-2 to solve the next tooth profile point; if it does not satisfy, it means that all tooth profile points of the current section have been solved, and execute step 5-8. Step 5-8: Determine if all cross sections have been traversed: Let the cross section index i = i + 1, and determine if i satisfies i ≤ M: If it does, it means there are still cross sections that have not been solved, return to step 5-2, and solve the tooth profile points of the next cross section; if it does not, it means that all cross sections have been solved, and proceed to step 5-9. Steps 5-9: Once all tooth surface point cloud data has been solved, the program ends.

[0016] Compared with the prior art, the present invention has the following significant advantages: (1) The proposed method has a simple invention principle and stable solution, which solves the problem of numerical solution being non-target solution often encountered in the process of numerical solution of point cloud of non-involute variable thickness gear tooth surface; (2) The proposed method proposes a method for numerical solution of initial value of non-involute variable thickness gear, which can significantly reduce the number of iterations required for convergence and reduce the occupation of computing resources; (3) When solving point cloud of non-involute variable thickness gear, the proposed method obtains the point cloud directly from the tooth surface equation through numerical solution, without cumulative error, and the calculation result has high accuracy. Attached Figure Description

[0017] Appendix Figure 1This is a schematic diagram illustrating the creation of a coordinate system for a non-involute variable-thickness gear with intersecting axes undergoing spatial transformation. Figure 1 (a) corresponds to the involute thickening gear per revolution Rotation Schematic diagram Figure 1 In section (b), the axial intersection angle of the corresponding intersecting shaft variable thickness gear pair is... Schematic diagram Figure 1 (c) corresponds to a non-involute variable thickness gear around which the gear is wrapped. Rotation Schematic diagram.

[0018] Appendix Figure 2 This is a schematic diagram illustrating the creation of a coordinate system for the spatial transformation of an interlaced, non-involute, variable-thickness gear. (Attached) Figure 2 (a) corresponds to the involute thickening gear per revolution Rotation Rotational speed of the gear around its axis A schematic diagram, Figure 2 (b) corresponds to the non-involute variable thickness gear surrounding the center. Rotation Rotational speed of the gear around its axis A schematic diagram, Figure 2 In the middle (c), the crossover angle corresponding to the cross-shaft variable thickness gear pair is: A schematic diagram.

[0019] Appendix Figure 3 This is a flowchart of the direct method for solving the point cloud of the tooth surface of a non-involute variable-thickness gear in the existing technology.

[0020] Appendix Figure 4 This is a schematic diagram of the indirect method for obtaining point clouds of non-involute tooth surfaces in existing technologies.

[0021] Appendix Figure 5 This is a schematic diagram illustrating the multiple solutions for solving the point cloud of a non-involute variable-thickness gear according to the present invention.

[0022] Appendix Figure 6 This is a flowchart of the solution process for the unknown variable of the addendum circle of a non-involute variable thickness gear in this invention.

[0023] Appendix Figure 7 This is a schematic diagram showing the relationship between the development angles of the involute thickened gear and the non-involute thickened gear in this invention.

[0024] Appendix Figure 8 This is a flowchart of the solution process for the unknown variables in the intermediate tooth profile of a non-involute variable-thickness gear in this invention.

[0025] Appendix Figure 9 This is a flowchart of the point cloud solution for non-involute variable thickness gears in this invention.

[0026] Appendix Figure 10This is a schematic diagram of the point cloud of the tooth surface of a non-involute thickened gear in an embodiment of the present invention.

[0027] Appendix Figure 11 This is a comparison of the number of iterations of the method proposed in this invention with the traditional process in the embodiments of this invention. Figure 11 In the middle (a), the number of iterations required to solve the point cloud using the traditional process is represented. Figure 11 In (b), the number of iterations required to solve the point cloud using the method proposed in this invention is represented. Detailed Implementation

[0028] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0029] The problem of finding multiple solutions to the point cloud of the tooth surface of a non-involute variable-thickness gear using the direct method: Let the equation of the tooth surface of the involute gear be: : (3), where It is the tooth surface equation On the coordinate axes The amount on, It is the involute development angle of the involute-thickening gear. It is the rotation angle of the involute gear with increasing thickness. It is the starting angle of the involute thickening gear.

[0030] According to the conjugate principle, the involute development angle, rotation angle, and starting angle of involute-variant gears and non-involute-variant gears have a one-to-one correspondence. Therefore, the tooth surface equation of the non-involute-variant gear can be written as: : (4), where It is the tooth surface equation On the coordinate axes The amount on, The contact line equation is for a non-involute variable thickness gear. From equation (4), it can be seen that the tooth surface equation of a non-involute variable thickness gear contains 3 unknowns. Solving for any point on the tooth surface of a non-involute variable thickness gear requires constructing 3 equations. Based on the positional relationship, the third-order nonlinear equation system for solving the point cloud of the tooth surface of a non-involute variable thickness gear is constructed as shown in equation (5): (5), in the formula Indicates the non-involute variable thickness gear. On the first cross section The coordinates of each tooth profile point It is the radius of the circle corresponding to a point on the tooth surface to be solved. It is the z-axis coordinate value corresponding to the cross section of the gear at a certain point on the tooth surface.

[0031] In use Figure 3 When solving point clouds in a process, the numerical solution obtained through iteration is quite sensitive to the given initial values. For example... Figure 5 As shown, taking the determination of the coordinates of the right tooth surface of a non-involute variable thickness gear as an example, two involutes can be drawn from a point on the base circle of the involute variable thickness gear. Involute one is counterclockwise, and involute two is clockwise. The radius of the non-involute variable thickness gear... The circle intersects the two involutes at four points, representing four possible scenarios when using the quasi-Newton method to solve the point cloud of the tooth surface of a non-involute variable-thickness gear: The red tooth profile segment located on the involute of the involute thickening gear has a positive development angle, which is the target solution. Located on the green extension line of the involute, outside the tooth surface, the corresponding development angle is positive, but smaller than... A larger unfolding angle indicates a non-target solution. Located on the clockwise involute 2, its unfolding angle is located at Near the opposite of the expansion angle value, outside the tooth surface, is a non-target solution; Located on the clockwise involute 2, its unfolding angle is located at Near the negative of the corresponding expansion angle, outside the tooth surface, is a non-target solution. Furthermore, when calculating the point cloud coordinates near the tooth root of a non-involute thickened gear, if the initial iteration value is inappropriate, convergence may fail within the predetermined number of iterations, leading to solution failure.

[0032] Taking the parameters of the non-involute variable thickness gear shown in Table 1 as an example, the coordinate data of the intersection point of the addendum cone and the tooth surface of the non-involute variable thickness gear are obtained using different initial values. Figure 5 The four scenarios are summarized in Table 1.

[0033] Table 1 Parameter Settings for Non-Involute Variable Thickness Gear Pairs

[0034] Table 2. Coordinates of point cloud at the same location on the tooth surface under different initial values.

[0035] Therefore, in order to obtain the accurate non-involute helical surface of the non-involute thickened gear, it is necessary to propose a criterion for determining the numerical solution of the unknown variable and to exclude non-target solutions. At the same time, in order to reduce the number of iterations required for convergence and reduce the possibility of iteration failure, the initial value should be given as close as possible to the target solution.

[0036] This example addresses the problem of multiple solutions encountered when using the direct method to obtain point clouds of non-involute variable thickness gear tooth surfaces. It proposes a criterion for determining the numerical solution and improves the solution process of the traditional direct method for obtaining point clouds of non-involute variable thickness gear tooth surfaces. The proposed method can guarantee that the numerical solution obtained through iteration is the target solution. This example also optimizes the initial value provisioning method, which can significantly reduce the number of iterations required for numerical solution and accelerate the convergence speed.

[0037] This example addresses two specific problems: first, it provides a criterion for determining numerical solutions, effectively distinguishing between the target solution and non-target solutions, ensuring that all numerical solutions obtained through iteration are the target solution; second, it provides a given scheme for initial values ​​during iteration, reducing the number of iterations required for convergence and accelerating the convergence speed. The specific implementation steps are as follows: The first step is to solve for the unknown variable data corresponding to the addendum circle of the non-involute variable thickness gear. remember Given the addendum circle radii of different cross-sections of an involute gear with varying thickness, then what is the involute development angle corresponding to the addendum circles of different cross-sections of the involute gear with varying thickness? It can be obtained from equation (6): (6), in the formula It is the pressure angle corresponding to the tooth tip circle with different cross-sections.

[0038] According to the definition of the involute development angle, its value cannot be less than 0. According to the gear meshing principle, for a pair of normally meshing gears, the addendum circles of the two gears must intersect; otherwise, the gears will disengage. Therefore, the involute development angle corresponding to the addendum circle of a non-involute variable thickness gear must be smaller than the involute development angle corresponding to a non-involute variable thickness gear. The relationship shown in equation (7) is satisfied: (7) The tangent of the pressure angle at the middle of the tooth profile of the involute variable thickness gear is used as the initial iterative value of the development angle corresponding to the tip circle of different sections of the non-involute variable thickness gear, denoted as Let the rotation angle corresponding to the middle of the tooth width of the involute gear be . , as the initial value for the iteration of the rotation angle corresponding to the addendum circle of different cross sections of the non-involute variable thickness gear; the initial value for the iteration of the starting angle is set to .remember The addendum circle radii are for different cross-sections of non-involute variable thickness gears. Given the axial position values ​​of different sections of a non-involute variable thickness gear, then the unknown variable corresponding to the addendum circle of the non-involute variable thickness gear is... , , Can be derived from Figure 6 The process shown is as follows.

[0039] Figure 6 The solution process shown contains two nested loops. The inner loop determines whether the numerical solution obtained by the quasi-Newton method is the target solution. If it is, it can be output directly; otherwise, it will be sorted by size. The step size is changed to change the initial value of the starting angle, and the solution is repeated. The outer loop is used to traverse and solve for the unknown variables of different cross sections. Each time the outer loop is completed, the position of the cross section needs to be changed. The value and the corresponding tooth tip circle radius The initial value of the position variable in each subsequent loop is the target solution obtained in the previous outer loop, except for the first outer loop.

[0040] The second step is to solve for the unknown variable data corresponding to different radii of the tooth profile at the intermediate section of the non-involute variable thickness gear: Based on the aforementioned tooth surface equations of involute and non-involute variable thickness gears, the tooth surfaces of involute and non-involute variable thickness gears are conjugate tooth surfaces. The tooth surface equation of the non-involute variable thickness gear is determined by the variables of the involute variable thickness gear itself. , and This is represented as follows. According to the gear meshing principle, when a non-involute thickened gear's tooth tip enters mesh, the meshing point is generally close to the base circle of the involute thickened gear. It should be the minimum value within the range of variation of the development angle of a non-involute variable-thickness gear. For example... Figure 7 As shown, let the development angle corresponding to the base circle of the involute thickened gear be . Since the tooth profile of the involute thickened gear is an involute and the base circle is the starting circle of the involute, the development angle corresponding to the base circle is 0. Let the development angle corresponding to the base circle of the non-involute thickened gear be ; Since the tooth profile height of the involute thickened gear and the non-involute thickened gear is similar, there exists a relationship as shown in equation (8): (8) Due to The value is always 0, therefore the development angle of the involute gear corresponding to the base circle of the non-involute thickening gear is: (9) In order to include the actual range of the development angle of non-involute variable thickness gears, the hypothetical maximum value of the development angle corresponding to different cross-sectional profiles of non-involute variable thickness gears is defined as follows: ,but Recommended: (10), where —Magnification factor, generally taken as 1.1-1.2.

[0041] Therefore, the range of the development angle corresponding to the tooth profile of a non-involute variable-thickness gear must be within the range shown in equation (11): (11), Let the radius of the circle containing each point on the tooth profile at the midpoint of the tooth width of the non-involute variable-thickness gear be . Let the axial position of the middle part of the tooth width of the non-involute variable-thickness gear be denoted as . The initial values ​​of the unknown variables are set in the same way as those set in the first step of calculating the addendum circle. Therefore, the unknown variables corresponding to different radii in the tooth profile at the middle of the non-involute variable tooth width are... , , Can be derived from Figure 8 The process shown yields, where, for The value corresponding to the cross section at the middle of the tooth width. Figure 8 The process shown is the same as Figure 6 The difference in the process shown is that, Figure 8 Each completion of the outer loop only requires changing the radius. The value will be used for the next outer loop iteration; Figure 6 It is not only necessary to change the radius value corresponding to the tooth tip circle The axial position of the cross section also needs to be changed. The parameter value.

[0042] The third step involves providing initial values ​​for the entire tooth surface based on Lagrange interpolation: Before solving the point cloud of the entire tooth surface of the non-involute variable-thickness gear, providing suitable initial values ​​for the numerical iteration of unknown variables will effectively accelerate the convergence speed, ensure convergence, and effectively prevent the numerical solution from converging to a non-target solution. Based on the aforementioned dataset of unknown variables corresponding to the addendum circle of the non-involute variable-thickness gear and the dataset of unknown variables corresponding to the tooth profile of the intermediate section, the initial values ​​for different radii of the tooth profile of the intermediate section are first determined. Lagrangian fitting is performed on the corresponding dataset of unknown variables; then, based on the tooth tip circle of each section... Based on the corresponding dataset of unknown variables and referring to the Lagrange equation for the intermediate section, the Lagrange equations for other cross sections are derived. The specific process is as follows.

[0043] First, the Lagrange equation is obtained by fitting the dataset of unknown variables corresponding to the tooth profile of the intermediate section of a non-involute variable-thickness gear: (12), where This is the one-dimensional Lagrange interpolation equation corresponding to the development angle of the tooth profile at the intermediate section. The one-dimensional Lagrange interpolation equation corresponding to the rotation angle of the tooth profile at the intermediate section is given. Let (12) be the one-dimensional Lagrange interpolation equation corresponding to the starting angle of the tooth profile at the intermediate section. Based on equation (12), the initial equations for the other sections can be obtained as follows: (13), where This is the two-dimensional Lagrange interpolation equation corresponding to the development angle of the tooth profile at the intermediate section. The equation for the two-dimensional Lagrange interpolation corresponding to the rotation angle of the tooth profile at the mid-section is given. The given equation is the two-dimensional Lagrange interpolation equation corresponding to the starting angle of the tooth profile at the intermediate section. It is the involute development angle of the involute thickening gear corresponding to the tooth tip circle of the middle section; It is the rotation angle of the involute gear with increasing thickness corresponding to the tooth tip circle of the middle section; It is the starting angle of the involute thickening gear corresponding to the tip circle of the intermediate cross section.

[0044] Step 4: Solving the point cloud of the tooth surface of the non-involute variable thickness gear: In this example, a three-layer nested loop is used to solve the point cloud of the tooth surface of the non-involute variable thickness gear. The inner loop is used to determine the rationality of the numerical solution. Equation (11) is used as the judgment condition. If the numerical solution is determined to be non-target solution, the initial value of the starting angle will be changed according to the step size and the numerical solution will be performed again. If the numerical solution is determined to be target solution, the inner loop will be exited and the solution of the next point will continue. The middle loop is used to solve the point cloud of the tooth profile. It is used to solve the point cloud of the tooth profile of different sections. After completing each step, the point cloud of the tooth profile is solved. The corresponding section on the th cross-section After solving for each tooth profile point, it needs to be changed once. The value is performed for the first time The first section Solving for each tooth profile point; the outer loop is used to change the cross section, when the first tooth profile point is completed. After solving for all tooth profile points on a cross section, the axial position parameters need to be changed. The value of the first The solution involves calculating the values ​​of all tooth profile points across a given cross section. After each inner loop iteration (i.e., after calculating the numerical values ​​of each tooth profile point), the 3D coordinate data of that point is exported. The specific process is as follows: Figure 9 As shown.

[0045] Instance verification Based on the stable solution method for point clouds of non-involute variable thickness gears proposed in this example, the point clouds of non-involute variable thickness gears under the parameters in Table 1 are as follows: Figure 10 As shown, no non-target solutions or non-convergence occurred during the solution process. The convergence was rapid and accurate, and the computer's computing resources were minimal.

[0046] Figure 3 The traditional process shown requires the following number of iterations to calculate the point cloud of the tooth surface of a non-involute variable-thickness gear: Figure 11 As shown in (a), the number of iterations required to calculate the point cloud of the tooth surface of a non-involute variable-thickness gear based on the method described in this invention is as follows: Figure 11(b) The traditional process requires an average of 11.61 iterations, and non-convergence occurs near the base circle at the small end of the non-involute thickened gear; the method requires an average of 3.15 iterations, does not fail to converge, and reduces the number of iterations by 72.87%, making the method stable and fast in convergence.

Claims

1. A method for modeling point clouds of the tooth surface of marine non-involute variable-thickness gears, characterized in that, Includes the following steps: Step 1: Establish the conjugate tooth surface equations for involute variable thickness gears and non-involute variable thickness gears, and construct a third-order nonlinear equation system for solving the point cloud of the tooth surface of non-involute variable thickness gears. Step 2: Solve for the unknown variable data corresponding to the addendum circle of the non-involute variable thickness gear; Step 3: Solve for the unknown variable data corresponding to different radii of the tooth profile at the intermediate section of the non-involute variable thickness gear; Step 4: Based on the Lagrange interpolation method, construct the iterative initial values ​​of the unknown variables of each section of the full tooth surface according to the datasets obtained in Step 2 and Step 3; Step 5: Based on the obtained initial values ​​of the iteration, use a three-level nested loop to solve the point cloud of the tooth surface of the non-involute variable thickness gear. In the process of solving, the range of the involute development angle of the non-involute variable thickness gear is used as the judgment condition for the numerical solution to exclude non-target solutions.

2. The method for modeling point clouds of marine non-involute variable thickness gear tooth surfaces according to claim 1, characterized in that, In step 2, record For involute gears with varying thicknesses, the addendum circle radii are given by different cross-sections, and the corresponding involute development angles are given by different cross-sections. From equation (6), we get: (6), in the formula This refers to the pressure angle corresponding to the addendum circle of different cross-sections. The involute development angle cannot be less than 0. According to the gear meshing principle, the involute development angle corresponding to the addendum circle of a non-involute variable thickness gear is necessarily smaller than the involute development angle corresponding to the involute variable thickness gear. That is, the involute development angle of a non-involute variable thickness gear... Satisfy the relationship shown in equation (7): (7) The tangent of the pressure angle at the middle of the tooth profile of the involute variable thickness gear is used as the initial iterative value of the development angle corresponding to the tip circle of different sections of the non-involute variable thickness gear, denoted as Let the rotation angle corresponding to the middle of the tooth width of the involute gear be . , as the initial value for the iteration of the rotation angle corresponding to the addendum circle of different cross sections of the non-involute variable thickness gear; the initial value for the iteration of the starting angle is set to ,remember The addendum circle radii are for different cross-sections of non-involute variable thickness gears. Given the axial position values ​​of different sections of a non-involute variable thickness gear, then the unknown variable corresponding to the addendum circle of the non-involute variable thickness gear is... , , The solution can be obtained as follows: The solution process includes two nested loops. The inner loop determines whether the numerical solution obtained by the quasi-Newton method is the target solution. If it is, it is output directly; otherwise, it is sorted by size. The step size changes the initial value of the starting angle, and the solution is repeated. The outer loop is used to traverse and solve for the unknown variables of different cross sections. Each time the outer loop is completed, the position of the cross section needs to be changed. The value and the corresponding tooth tip circle radius The initial value of the position variable in each subsequent loop is the target solution obtained in the previous outer loop, except for the first outer loop.

3. The method for modeling point clouds of marine non-involute variable thickness gear tooth surfaces according to claim 2, characterized in that, The specific solution process in step 2 is as follows: Step 2-1: Initialize the outer loop variable. Let the outer loop variable be i, which is used to represent different cross sections of the gear. Set the initial value to 0 and the value range is 0 ≤ i ≤ M, where M is the total number of cross sections. After solving each cross section, execute i = i + 1 until all cross sections are traversed. Step 2-2: Set the initial values ​​for the iteration of the unknown variables. For the current section i, set the initial values ​​for the iteration of the three unknown variables: the initial value of the involute development angle. ,in The initial value of the involute development angle corresponding to the current tooth tip circle; initial value of the starting angle. , The initial value of the starting angle corresponding to the middle of the tooth width; the initial value of the rotation angle. Initially set to ; Step 2-3: Initialize the inner loop variable k. Let the inner loop variable be k, which is used to adjust the initial value of the starting angle. The initial value is set to k = 0, where k is an integer and 0 ≤ k ≤ n, and n is the preset maximum number of adjustments. At the same time, reset the current initial value of the starting angle. According to the formula Perform initial settings; Steps 2-4: Solve the nonlinear equation system using the quasi-Newton method, with the currently given initial values. The following third-order nonlinear equations are solved using the quasi-Newton method: Position Equation 1: , The radius of the tooth tip circle at the current cross-section. Position Equation 2: , This represents the axial position of the current cross-section. Conjugate meshing equation: Solve for the unknown variable value corresponding to the current numerical solution. ; Step 2-5: Determine if the numerical solution is the target solution, and check the involute development angle in the numerical solution obtained in Step 2-4. Does the objective solution meet the criteria? ,in Let be the development angle of the tip circle corresponding to the current section of the involute thickened gear. If the above conditions are met, the numerical solution is the target solution, and we jump to step 2-7. If the above conditions are not met, the numerical solution is a non-target solution, and we execute step 2-6. Steps 2-6: Adjust the initial value of the starting angle and solve again: Let the inner loop variable k = k + 1, and then... Update the initial angle value, that is, change it by a fixed step size of 2π / n. Then return to steps 2-4 and use the quasi-Newton method again with the new initial angle value; Step 2-7: Save the target solution for the current cross-section: Save the current numerical solution as the target solution corresponding to the tooth tip circle of the cross-section. ； Step 2-8: Update the initial value of the position variable for the next iteration: To speed up convergence, the solution obtained from the current section is used as the initial value of the position variable for the next section, i.e., let: ； Step 2-9: Determine if all cross sections have been traversed. Let the outer loop variable i = i + 1, and check if i satisfies i≤ M: If it does, it means there are still cross sections that have not been solved, return to step 2-2, and solve the unknown variable of the tooth tip circle of the next cross section; if it does not satisfy, it means that the unknown variable of the tooth tip circle of all cross sections has been solved, and proceed to the next step. Step 2-10: The solution for the unknown variable data corresponding to the tooth tip circle is complete, and the process ends.

4. The point cloud modeling method for the tooth surface of marine non-involute variable thickness gears according to claim 3, characterized in that, In step 3, based on the tooth surface equations of the involute-thickness gear and the non-involute-thickness gear, the tooth surfaces of the involute-thickness gear and the non-involute-thickness gear are conjugate tooth surfaces. The tooth surface equation of the non-involute-thickness gear is determined by the variables of the involute-thickness gear itself. , and To illustrate, according to the gear meshing principle, when the tooth tip of a non-involute thickened gear engages, the meshing point is close to the base circle of the involute thickened gear. This should be the minimum value within the range of variation of the development angle of a non-involute variable thickness gear, denoted as the development angle corresponding to the base circle of the involute variable thickness gear. Since the tooth profile of an involute-thickened gear is involute and its base circle is the involute starting circle, the development angle corresponding to the base circle is 0. Let the development angle corresponding to the base circle of a non-involute-thickened gear be... Since the involute thickened gear and the non-involute thickened gear have similar tooth profile heights, the relationship shown in equation (8) exists: (8) Due to The value is always 0, therefore the development angle of the involute gear corresponding to the base circle of the non-involute thickening gear is: (9) In order to include the actual range of the development angle of non-involute variable thickness gears, the hypothetical maximum value of the development angle corresponding to different cross-sectional profiles of non-involute variable thickness gears is defined as follows: ,but Recommended: (10), where This is the magnification factor, ranging from 1.1 to 1.

2. For non-involute thickened gears, the range of the development angle corresponding to the tooth profile is within the range shown in equation (11): （11）， Let the radius of the circle containing each point on the tooth profile at the midpoint of the tooth width of the non-involute variable-thickness gear be . Let the axial position of the middle part of the tooth width of the non-involute variable-thickness gear be denoted as . If the initial values ​​of the unknown variables are set in the same way as those set in the first step of calculating the addendum circle, then the unknown variables corresponding to different radii of the tooth profile at the middle of the tooth width of the non-involute variable-thickness gear are... , The solution can be obtained, and the solution process differs from that in step 2 in that only the radius needs to be changed after each completion of the outer loop. The value is used to proceed to the next outer loop, where is the value corresponding to the cross section at the middle of the tooth width.

5. The method for modeling point clouds of marine non-involute variable thickness gear tooth surfaces according to claim 4, characterized in that, Step 4 specifically involves: Based on the aforementioned datasets of unknown variables corresponding to the addendum circle of the non-involute variable-thickness gear and the datasets of unknown variables corresponding to the tooth profile of the intermediate section, firstly, Lagrange fitting is performed on the datasets of unknown variables corresponding to different radii of the tooth profile of the intermediate section; then, based on the datasets of unknown variables corresponding to the addendum circle of each section and referring to the Lagrange equation of the intermediate section, the Lagrange equations for other cross sections are given. The specific process is as follows: First, the Lagrange equation is obtained by fitting the dataset of unknown variables corresponding to the tooth profile of the intermediate section of the non-involute variable-thickness gear: （12）， Based on equation (12), the initial value equations for the other sections can be obtained as follows: （13）， In the formula It is the involute development angle of the involute thickening gear corresponding to the tooth tip circle of the middle section; It is the rotation angle of the involute gear with increasing thickness corresponding to the tooth tip circle of the middle section; It is the starting angle of the involute thickening gear corresponding to the tip circle of the intermediate cross section.

6. The method for modeling point clouds of marine non-involute variable thickness gear tooth surfaces according to claim 5, characterized in that, Step 5 is specifically: use a three-layer nested loop to solve the point cloud of the tooth surface of the non-involute variable thickness gear. The inner loop is used to determine the rationality of the numerical solution. Equation (11) is used as the judgment condition. If the numerical solution is determined to be non-target solution, the initial value of the starting angle will be changed according to the step size and the numerical solution will be performed again. If the numerical solution is determined to be target solution, the inner loop will be exited and the solution of the next point will be performed. The intermediate layer iteratively solves the point cloud of the tooth profile, used to solve the point cloud of the tooth profile at different cross-sections. Each time the corresponding point cloud at the i-th cross-section is completed... After solving for each tooth profile point, it needs to be changed once. The value is performed for the first time The first section Solving for each tooth profile point; the outer loop is used to change the cross section, when the first tooth profile point is completed. After solving for all tooth profile points on a cross section, the axial position parameters need to be changed. The value is used to solve for all tooth profile points of the i-th section; after each inner loop is completed, that is, after each tooth profile point is solved, the 3D coordinate data of the tooth profile point is exported.

7. The method for modeling point clouds of marine non-involute variable thickness gear tooth surfaces according to claim 6, characterized in that, The specific steps of step 5 are as follows: Step 5-1: Initialize the loop variables. Let the outer loop variable be i, which represents different cross sections of the gear. Its value range is 0 ≤ i ≤ M, where M is the total number of cross sections. Let the middle loop variable be j, which represents each point on the tooth profile on the same cross section. Its value range is 0 ≤ j ≤ N, where N is the total number of tooth profile points on a single cross section. Let the inner loop variable be k, which represents the number of iterations. The initial value is set to k=0, where k is an integer and 0 ≤ k ≤ n. Step 5-2: Given the initial values ​​of the unknown variables, calculate and assign the initial values ​​of the unknown variables based on the pre-constructed Lagrange interpolation function, using the current section index i and tooth profile point index j: Initial value of the involute development angle. Initial value of the starting angle Initial value of rotation angle ; Step 5-3: Solve the nonlinear equation system using the quasi-Newton method. With a given initial value, solve the third-order nonlinear iterative equation system of the point cloud of the non-involute variable-thickness gear tooth surface using the quasi-Newton method. The specific equation system is as follows: Position Equation 1: ; Position Equation 2: ; Conjugate meshing equation: ; Solve for the unknown variable value corresponding to the current numerical solution. ; Step 5-4: Determine if the numerical solution is the target solution, and determine the involute development angle in the obtained numerical solution. Does the objective solution meet the criteria? ,in The maximum value of the development angle of the non-involute variable-thickness gear in the current section is assumed. If the above conditions are met, the numerical solution is the target solution, and the process jumps to step 5-6; if the above conditions are not met, the numerical solution is a non-target solution, and step 5-5 is executed. Step 5-5: Adjust the initial value of the starting angle and solve again: Let the inner loop variable k = k + 1, change the initial value of the starting angle according to the set step size, and then return to step 5-3 to solve the nonlinear equation system again using the quasi-Newton method with the new initial value of the starting angle. Steps 5-6: Output the target solution and 3D coordinates of the current tooth profile point: Save the current numerical solution as the target solution for this tooth profile point and record the target solution: Simultaneously calculate and output the three-dimensional coordinate data of that point: X coordinate: , Y coordinate: , Z-coordinate: ; Step 5-7: Determine whether the tooth profile points of the current section have been traversed. Let the tooth profile point index j = j + 1, and determine whether j satisfies j ≤ N. If it does, it means that there are still tooth profile points in the current section that have not been solved. Return to step 5-2 to solve the next tooth profile point. If the condition is not met, it means that all tooth profile points of the current section have been solved, and proceed to step 5-8; Step 5-8: Determine if all cross sections have been traversed: Let the cross section index i = i + 1, and determine if i satisfies i ≤ M: If it satisfies, it means that there are still cross sections that have not been solved, return to step 5-2, and solve the tooth profile points of the next cross section; if it does not satisfy, it means that all cross sections have been solved, and execute step 5-9. Steps 5-9: Once all tooth surface point cloud data has been solved, the program ends.

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