Numerical simulation method for double-meshing measurement of involute cylindrical gear

Abstract

The invention discloses a numerical simulation method for double-meshing measurement of an involute cylindrical gear, belongs to the field of precision test technologies and instruments, and relates to a rolling measurement method for a Matte gear and a measured gear in a tight meshing state. In the rolling process of the gear, the quality of the gear is evaluated by measuring and recording the tiny change of the center distance. According to the method, a mathematical model of a Matter gear and a measured gear containing various errors is established according to a gear meshing theory. According to the method, the correlation model between the radial comprehensive deviation and the tooth profile deviation is constructed, and the problem of solving the tooth profile meshing point equation is converted into the problem of solving the tangent point of the two involute tooth profiles, so that the calculation complexity of the algorithm is reduced, and the calculation precision is improved.

Description

Technical Field

[0001] The present invention belongs to the field of precision testing technology and instruments, and in particular relates to an error simulation analysis method based on a double-flank meshing mathematical model of an involute cylindrical gear. Background Art

[0002] Gear transmissions are widely used in a variety of fields, including aerospace, instrumentation, automotive, and precision machinery. Gear accuracy and quality directly impact the efficiency, noise, motion accuracy, and service life of machinery and equipment. Therefore, gear inspection is particularly important.

[0003] There are many instruments for inspecting gears, including single-flank gear meshing testers, double-flank gear meshing testers, and gear measuring centers. For small and medium-module gears, double-flank gear meshing testers are often used, as they offer advantages such as low environmental requirements, simple structure, and easy operation.

[0004] Current double-mesh simulation technology suffers from high computational complexity and insufficient accuracy. Traditional methods rely on defining the tooth surface's normal unit vector to determine the contact point. This model first considers tooth-to-tooth contact (when the normal vectors match) and then checks for tooth tip-to-tooth contact (when the normal vectors don't match). This approach significantly increases computational costs.

[0005] The present invention transforms the problem of solving the tooth profile meshing point equation into the problem of solving the tangent point of two involute tooth profiles, thereby reducing the computational complexity of the algorithm and improving the computational accuracy. Summary of the Invention

[0006] The technical objective of this invention is to propose a double-mesh gear measurement method. Specifically, this method involves rolling a matt gear in close mesh with the gear being measured. During the rolling process, the gear quality is assessed by measuring and recording small changes in the center distance. Based on gear meshing theory, this method establishes mathematical models of the matt gear and the gear being measured, including various errors.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is a double-flank meshing measurement simulation method for involute cylindrical gears based on the meshing principle. The implementation process of this method is as follows:

[0008] Double-mesh measurement involves rolling a matt gear in close mesh with the gear being tested. During this rolling process, minute changes in the center distance are measured and recorded to assess gear quality. This method first establishes mathematical models of the matt gear and the gear being tested, including various errors, based on gear mesh theory.

[0009] The implementation process of this method is as follows:

[0010] (1) In the Cartesian coordinate system, the involute expression of a tooth profile of a cylindrical gear is listed. Then, by changing the coordinates, the expressions of all the involute tooth profiles of the cylindrical gear are obtained, and then the mathematical model of the cylindrical gear is obtained.

[0011] (2) Determine the contact point of the working side tooth profile based on the mathematical model of the cylindrical gear. The problem of solving the equation that the velocity vector of the involute tooth profile at the contact point, i.e. the relative motion, should be perpendicular to the normal vector of the tooth profile, is transformed into: Under the condition of rotating the gear tooth profile Solve and measure the tooth profile tangent point.

[0012] (3) Determine the engagement angle The numerical model uses the gear tooth profile point cloud data in the absolute coordinate system as input to calculate the contact points between the matt gear and the working tooth profile of the gear being tested (the non-working tooth profile is not included in the contact analysis). The first point of the gear being tested is obtained through geometric solution. tooth profile With Matt Gear tooth profile Meshing angle , so that the gear tooth profile is With Matt Wheel Tooth Profile Meshing.

[0013] (4) Determine the optimal solution of the center distance C. In order to determine the correct center distance in the tight meshing state, the center distance (C min , C max ) Interval solution C target . Objective function Defined as the tolerance at the non-working side tooth profile (where >0 gap or <0 penetration). If the objective function The absolute value is less than the error tolerance , it is considered as converged output C target Error tolerance The selection of the value of and the determination of the degree of tooth profile discretization need to be balanced between calculation accuracy and operation efficiency according to the specific application scenario.

[0014] (5) Calculate the radial comprehensive deviation. Rotate the gear to be measured And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the rotation angle of each gear being measured. The corresponding center distance Target solution: When the gear under test rotates one circle, a complete radial comprehensive deviation curve can be obtained.

[0015] Furthermore, in the Cartesian coordinate system, the expression of the involute tooth profile of the Matt gear is solved:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] Where, is the base circle radius of the Matt gear, 、 Respectively characterize 、 Tooth profile involute position parameters. The starting position of the involute tooth profile is determined by the parameters and Determined. Based on the single tooth model expansion Number of teeth, i=1, 2, 3, ..., , build a complete gear model. Coordinate parameters 、 and 、 In the example, the subscript M represents the coordinate system, and the superscript 、 Corresponding to the non-working side and working side tooth profiles of Matt gear respectively. Determines the tooth width of the base circle of the matt gear.

[0022]

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] Furthermore, for the gear being tested, and Respectively represent the base circle radius of the tooth profile on the non-working side and the working side, 、 The starting position of the involute tooth profile is determined by the parameters 、 OK, based on the single tooth model expansion The number of teeth is used to build a full tooth numerical model. Coordinate parameters 、 and 、 In the example, subscript T represents the coordinate system, and superscript 、 They correspond to the non-working side and working side tooth profiles of the gear being tested. 、 Determines the base circle tooth width of the gear being measured.

[0029] Furthermore, based on the established gear involute mathematical model, a double-flank meshing model of the gear pair is established. The process is as follows:

[0030] In the double-meshing measurement gear pair meshing model, and are the reference coordinate systems of the Matt gear ΣM and the gear under test ΣT respectively. C represents the center distance variable, and The corresponding rotation angles are respectively the Matt gear ΣM and the gear being tested ΣT. When using the tooth contact analysis TCA method to simulate the meshing of two tooth surfaces of a gear pair, it is necessary to use a unified coordinate system. The mathematical description of the tooth surface position vectors of the Matt gear ΣM and the measured gear ΣT is established:

[0031]

[0032]

[0033]

[0034]

[0035] The full tooth numerical model uses the gear tooth profile point cloud data in the absolute coordinate system as input to calculate the contact point between the matt gear and the working tooth profile of the gear being tested. Axis-symmetrical distribution. Given the initial center distance C0, solve to obtain the gear under test tooth profile With Matt Gear tooth profile Meshing angle , its mathematical expression is:

[0036] , in, is the pitch circle radius of the gear being measured

[0037]

[0038]

[0039]

[0040]

[0041] Furthermore, in order to determine the optimal center distance in double-sided close meshing, a solution framework based on the bisection method is constructed, with the non-working tooth profile tolerance That is, the maximum penetration or minimum gap value is used as the objective function. The process is as follows: First, determine the meshing point angle , so that the tooth profile of the gear being measured With Matt Wheel Tooth Profile Meshing, determine the tolerance of the non-working tooth profile at this time , the calculation method is the same as the meshing point angle If penetration occurs, the objective function takes a negative value, and if there is a gap, the objective function takes a positive value. When the absolute value of the objective function meets the error tolerance When the current center distance The algorithm is terminated when it is judged to be the target solution; to ensure that the target center is Interval convergence of the boundary parameters and They are determined by the following constraint equations:

[0042]

[0043]

[0044]

[0045] Furthermore, the interval is narrowed down by bisection iteration to approximate the solution. After obtaining the optimal solution C, the angle of the gear being measured is rotated And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the center distance corresponding to each measured gear angle Target solution: a complete radial comprehensive deviation curve is obtained when the gear under test rotates one circle.

[0046] The method of the present invention aims to construct a correlation model between radial composite deviation and tooth profile deviation. Simultaneously, the method transforms the problem of solving the tooth profile meshing point equation into the problem of solving the tangent point of two involute tooth profiles, thereby reducing the algorithm's computational complexity and improving its accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a schematic diagram of the double-meshing measurement structure of the present invention.

[0048] Figure 2Schematic diagram of the involute tooth profile of the Matt gear and the gear being tested

[0049] Figure 3 Shown is a schematic diagram of the double-meshing measuring gear pair model

[0050] Figure 4 Schematic diagram of the working tooth profile calculation range.

[0051] Figure 5 The meshing angle is calculated Schematic diagram.

[0052] Figure 6 Is the tolerance Schematic diagram.

[0053] Figure 7 Is the tolerance Schematic diagram.

[0054] Figure 8 is a schematic diagram of the target center distance.

[0055] Figure 9 Center distance variation measured for two error-free gears in a double mesh.

[0056] Figure 10 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the tested gear is 19.5° and the pressure angle of the right tooth profile is 21°.

[0057] Figure 11 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the measured gear is 19° and the pressure angle of the right tooth profile is 20°.

[0058] Figure 12 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the measured gear is 19° and the pressure angle of the right tooth profile is 19°.

[0059] Figure 13 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the tested gear is 19°, the pressure angle of the right tooth profile is 19°, and the eccentricity is 0.1mm. DETAILED DESCRIPTION

[0060] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0061] The double-flank gear rolling test involves rolling a matt gear against the gear under test in close mesh. During the rolling process, minute changes in the center distance are measured and recorded to assess gear quality. This method first establishes mathematical models of the matt gear and the gear under test containing various errors based on gear meshing theory.

[0062] The implementation process of this method is as follows:

[0063] (1) In the Cartesian coordinate system, the involute expression of a tooth profile of a cylindrical gear is listed. Then, by changing the coordinates, the expressions of all the involute tooth profiles of the cylindrical gear are obtained, and then the mathematical model of the cylindrical gear is obtained.

[0064] (2) Determine the working side tooth profile, i.e., the contact point of the working surface. The problem of solving the equation that the velocity vector of the involute tooth profile at the contact point, i.e., the relative motion, should be perpendicular to the normal vector of the tooth profile, is transformed into: under the condition that the initial value of the center distance between the two cylindrical gears is known, the tooth profile of the rotating gear is obtained by rotating the gear. Solve and measure the tooth profile This approach allows for universal meshing characteristics research for any tooth shape without requiring special calculations of tooth profile singularities.

[0065] (3) Determine the engagement angle The numerical model uses the gear tooth profile point cloud data in the absolute coordinate system as input to calculate the contact points between the matt gear and the working tooth profile of the gear being tested (the non-working tooth profile is not included in the contact analysis). The first point of the gear being tested is obtained through geometric solution. tooth profile With Matt Gear tooth profile Meshing angle , so that the gear tooth profile is With Matt Wheel Tooth Profile Meshing.

[0066] (4) Determine the optimal solution of the center distance C. In order to determine the correct center distance in the tight meshing state, the center distance (C min , C max ) Interval solution C target The objective function f(C) is defined as the tolerance of the non-working tooth profile (non-working tooth profile) If the absolute value of the objective function f(C) is less than the error tolerance , it is considered as converged output C target Tolerance The selection of the value of and the determination of the degree of tooth profile discretization need to be balanced between calculation accuracy and operation efficiency according to the specific application scenario.

[0067] (5) Calculate the radial comprehensive deviation. Rotate the gear to be measured And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the rotation angle of each gear being measured. The corresponding center distance , when the gear under test rotates one circle, a complete radial comprehensive deviation curve is obtained.

[0068] Furthermore, in the Cartesian coordinate system, the expression of the involute tooth profile of the Matt gear is solved:

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] Where, is the base circle radius of the Matt gear, 、 Respectively characterize 、 Tooth profile involute position parameters. The starting position of the involute tooth profile is determined by the parameters and Determined. Based on the single tooth model expansion Number of teeth (i=1, 2, 3, ..., ), a complete gear model can be constructed (such as Figure 3 Coordinate parameters 、 and 、 In the example, the subscript M represents the coordinate system, and the superscript 、 Corresponding to the non-working side and working side tooth profiles of Matt gear respectively. Determines the tooth width of the Matt gear base circle.

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081] For the gear being tested, and Respectively represent the base circle radius of the tooth profile on the non-working side and the working side, 、 The starting position of the involute tooth profile is determined by the parameters 、 OK, based on the single tooth model expansion Number of teeth (i=1, 2, 3, ..., ), a full tooth numerical model can be established (such as Figure 3 Coordinate parameters 、 and 、 In the example, subscript T represents the coordinate system, and superscript 、 They correspond to the non-working side and working side tooth profiles of the gear being tested. 、 Determines the base circle tooth width of the gear being measured.

[0082] Based on the established gear involute mathematical model, a double-sided meshing model of gear pairs is established. The process is as follows:

[0083] Figure 3 The figure shows the meshing model of the double-meshing rolling measurement gear pair. and are the reference coordinate systems of the Matt gear ΣM and the gear under test ΣT respectively. C represents the center distance variable, and The corresponding rotation angles are the Matt gear ΣM and the gear being tested ΣT. When using the tooth contact analysis (TCA) method to simulate the meshing of two tooth surfaces of a gear pair, it is necessary to use a unified coordinate system. The mathematical description of the tooth surface position vectors of the Matt gear ΣM and the measured gear ΣT is established:

[0084]

[0085]

[0086]

[0087] The numerical model uses the gear tooth profile point cloud data in the absolute coordinate system as input to calculate the contact points between the matt gear and the working tooth profile of the gear being tested (the non-working tooth profile is not included in the contact analysis). Axisymmetric distribution (see Figure 4 Given the initial center distance C0, solve to obtain the gear tooth profile With Matt Gear tooth profile Meshing angle , its mathematical expression is:

[0088] , in, is the pitch circle radius of the gear being measured

[0089]

[0090]

[0091]

[0092]

[0093] In order to determine the optimal center distance in double-sided close meshing, a solution framework based on the bisection method is constructed with the non-working tooth profile tolerance (i.e. maximum penetration or minimum gap value) as the objective function The algorithm flow is as follows: First determine the meshing point angle , so that the tooth profile of the gear being measured With Matt Wheel Tooth Profile Meshing, determine the tolerance of the non-working tooth profile at this time (Calculation method is the same as meshing point angle If penetration occurs, the objective function takes a negative value, and if there is a gap, the objective function takes a positive value. When the current center distance The algorithm is terminated when it is judged to be the target solution; to ensure that the target center is Interval convergence, boundary parameters and They are determined by the following constraint equations:

[0094]

[0095]

[0096]

[0097] The approximate solution is obtained by iteratively narrowing the interval through bisection. After obtaining the optimal solution C, the gear being tested is rotated. And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the center distance corresponding to each measured gear angle , when the gear under test rotates one circle, a complete radial comprehensive deviation curve is obtained.

[0098] This invention aims to establish a correlation model between radial composite deviation and tooth profile deviation. Furthermore, the present invention transforms the problem of solving the tooth profile meshing point equation into the problem of solving the tangent point of two involute tooth profiles, thereby reducing the algorithm's computational complexity and improving its accuracy. The computer simulation program developed in this paper can simulate double-flank meshing to measure radial composite deviation. Table 1 lists the basic parameters of the Matt gear and the gear being measured.

[0099] Table 1 Main design parameters of Matt gear and tested gear

[0100]

[0101] To validate the effectiveness of the double-flank meshing numerical model, the first measurement must be performed with two error-free gears (matt gears) in close mesh. According to involute theory, the deviation from the theoretical center distance should be zero. However, due to tolerances, slight deviations from the theoretical center distance are expected within the permissible tolerance range. Based on this, the algorithm also simulates the radial composite deviation curve when the measured gears have pressure angle errors and eccentricity.

[0102] Figure 9 Center distance variation measured for two error-free gears in a double mesh.

[0103] Figure 10 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the tested gear is 19.5° and the pressure angle of the right tooth profile is 21°.

[0104] Figure 11 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the measured gear is 19° and the pressure angle of the right tooth profile is 20°.

[0105] Figure 12 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the measured gear is 19° and the pressure angle of the right tooth profile is 19°.

[0106] Figure 13 It is the radial comprehensive deviation when the pressure angle of the left tooth profile of the tested gear is 19°, the pressure angle of the right tooth profile is 19°, and the eccentricity is 0.1mm.

Claims

1. A numerical simulation method for double-meshing measurement of involute cylindrical gears, characterized in that: The implementation process of this method is as follows: (1) In the Cartesian coordinate system, the involute expression of a tooth profile of a cylindrical gear is listed. Then, by changing the coordinates, the expressions of all the involute tooth profiles of the cylindrical gear are obtained, and then the mathematical model of the cylindrical gear is obtained; (2) Determine the contact point of the working side tooth profile based on the mathematical model of the cylindrical gear; transform the equation of the involute tooth profile at the contact point, i.e. the velocity vector of the relative motion should be perpendicular to the normal vector of the tooth profile, into the equation of the working side ... Under the condition of rotating the gear tooth profile Solve and measure the tooth profile The tangent point; (3) Determine the engagement angle The numerical model uses the gear tooth profile point cloud data in the absolute coordinate system as input to calculate the contact points between the matt gear and the working tooth profile of the gear being tested (the non-working tooth profile is not included in the contact analysis). The first point of the gear being tested is obtained through geometric solution. tooth profile With Matt Gear tooth profile Meshing angle , so that the gear tooth profile is With Matt Wheel Tooth Profile Meshing; (4) Determine the optimal solution of the center distance C; To determine the correct center distance in the tight meshing state, based on the dichotomy method, the center distance (C min , C max ) Interval solution C target ; Objective function Defined as the tolerance at the non-working side tooth profile (where >0 gap or <0 penetration); if the objective function The absolute value is less than the error tolerance , it is considered as converged output C target ; (5) Calculate the radial comprehensive deviation; rotate the gear to be measured And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the rotation angle of each gear being measured. The corresponding center distance Target solution: When the gear under test rotates one circle, a complete radial comprehensive deviation curve can be obtained.

2. The numerical simulation method for double meshing measurement of involute cylindrical gears according to claim 1, characterized in that: In the Cartesian coordinate system, the expression of the involute tooth profile of the Matt gear is solved: Where, is the base circle radius of the Matt gear, 、 Respectively characterize 、 Tooth profile involute position parameters; the starting position of the involute tooth profile is determined by the parameters and Determine; Expand based on single tooth model Number of teeth, i=1, 2, 3, ..., , build a complete gear model; coordinate parameters 、 and 、 In the example, the subscript M represents the coordinate system, and the superscript 、 Corresponding to the non-working side and working side tooth profiles of Matt gear respectively; parameters Determine the tooth groove width of the base circle of the Matt gear; 。 3. The numerical simulation method for double meshing measurement of involute cylindrical gears according to claim 1, characterized in that: For the gear being tested, and Respectively represent the base circle radius of the tooth profile on the non-working side and the working side, 、 is the involute parameter of the corresponding tooth profile; the starting position of the involute tooth profile is determined by the parameter 、 OK, based on the single tooth model expansion The number of teeth is used to establish a full tooth numerical model; coordinate parameters 、 and 、 In the example, subscript T represents the coordinate system, and superscript 、 Corresponding to the non-working side and working side tooth profiles of the gear being tested; parameters 、 Determines the base circle tooth width of the gear being measured.

4. The numerical simulation method for double meshing measurement of involute cylindrical gears according to claim 1, characterized in that: Based on the established gear involute mathematical model, a double-sided meshing model of gear pairs is established. The process is as follows: In the double-meshing measurement gear pair meshing model, and are the reference coordinate systems of the Matt gear ΣM and the gear being measured ΣT respectively; C represents the center distance variable, and The corresponding rotation angles are respectively the Matt gear ΣM and the gear being tested ΣT; when the tooth contact analysis TCA method is used to simulate the meshing of the double tooth surfaces of the gear pair, it is necessary to use the unified coordinate system In the figure, a mathematical description of the tooth surface position vectors of the matt gear ΣM and the measured gear ΣT is established: The full-tooth numerical model takes the gear tooth profile point cloud data in the absolute coordinate system as input and calculates the contact points between the working tooth profile of the matt gear and the gear being measured; At the initial stage of simulation, the first tooth of the Matt gear and the gear under test are Axis-symmetrical distribution; given the initial center distance C0, solve to obtain the gear under test tooth profile With Matt Gear tooth profile Meshing angle , its mathematical expression is: , in, is the pitch radius of the gear being measured 。 5. The numerical simulation method for double meshing measurement of involute cylindrical gears according to claim 1, characterized in that: In order to determine the optimal center distance in double-sided close meshing, a solution framework based on the bisection method is constructed with the non-working tooth profile tolerance That is, the maximum penetration or minimum gap value is used as the objective function; the process is as follows: first determine the meshing point angle , so that the tooth profile of the gear being measured With Matt Wheel Tooth Profile Meshing, determine the tolerance of the non-working tooth profile at this time , the calculation method is the same as the meshing point angle Consistent; if penetration occurs, the objective function takes a negative value, if there is a gap, the objective function takes a positive value; when the absolute value of the objective function meets the error tolerance When the current center distance The algorithm is terminated when it is judged to be the target solution; to ensure that the target center is Interval convergence, boundary parameters and They are determined by the following constraint equations: 。 6. The numerical simulation method for double meshing measurement of involute cylindrical gears according to claim 1, characterized in that: The approximate solution is obtained by iteratively narrowing the interval through bisection method; after obtaining the optimal solution C, the angle of the gear being measured is rotated And repeat the meshing point angle Tolerance of tooth profile on non-working side The calculation is based on the dichotomy method to find the center distance corresponding to each measured gear angle Target solution: a complete radial comprehensive deviation curve is obtained when the gear under test rotates one circle.

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