A method for calculating meshing point of multi-tooth meshing surface gears based on tooth surface mathematical model

Abstract

The present invention belongs to a method for calculating the meshing point of multi-tooth meshing face gears for helicopters. Based on basic theories such as differential geometry and meshing principle, the meshing point of the tooth surface is further solved by constructing a mathematical model of the tooth surface of the face gear pair. In the process of solving the problem, the influence of multi-tooth meshing and tooth surface error is considered, so as to realize the rapid and accurate calculation of the meshing point of the tooth surface, and greatly improve the calculation accuracy of the meshing point. At the beginning of gear design, the tooth surface of the face gear can also be designed according to different working conditions. This method not only improves the design efficiency, but also greatly reduces the design cost, and at the same time provides technical support for the tooth surface design of high-performance transmission systems.

Description

Technical Field

[0001] The invention belongs to the technical field of tooth surface design of face gears for helicopters, and in particular is a method for calculating meshing points of multi-tooth meshing face gears based on a tooth surface mathematical model. Background Art

[0002] Face gears are widely used in aerospace, wind power transmission, vehicles and other fields due to their compact structure, high transmission efficiency and low sensitivity to installation errors. In this field, the complex working conditions of high speed and heavy load put forward higher requirements on the meshing performance of the meshing points of face gear tooth surfaces. For example, in many helicopter transmission system reports, NASA not only requires that the maximum contact stress at the meshing point be less than the yield strength of the material, but also requires that the tooth surface imprint (the contact ellipse expanded by the meshing point under load) account for 75%, and the transmission error (a comprehensive error at the meshing point) should be a high-order curve.

[0003] The accurate calculation of the tooth surface meshing point is a prerequisite for determining the maximum contact stress, tooth surface imprint and transmission error. It is of great significance for the meshing quality evaluation of face gear pairs and the design of face gears with high load-bearing contact performance. The determination of the traditional gear tooth meshing point is based on the principle of continuous tangential contact. When it comes to multi-tooth meshing, theoretically, the meshing behavior of the gear teeth shows obvious periodic characteristics. In the continuous meshing process, the positions of the tooth surface meshing points of different tooth pairs will show consistency. However, due to the tooth surface error caused by machining, after the tooth pair first enters the meshing, the contact of the subsequent tooth pairs participating in the meshing does not meet the principle of continuous tangency, and the actual meshing point of the subsequent meshing tooth pair will deviate from the theoretical meshing point. The existing technology lacks a numerical calculation method for the tooth surface meshing point that takes into account the tooth surface error. At present, the meshing point of the face gear pair is obtained through a gear test bench, which is used as feedback for the optimization design of the gear tooth surface, but the experimental method has a long cycle and huge design cost. Summary of the invention

[0004] The purpose of the present invention is to avoid the shortcomings of the prior art and provide a method for calculating the meshing point of multi-tooth meshing face gears based on a mathematical model of the tooth surface. Based on basic theories such as differential geometry and meshing principles, the influence of tooth surface errors on the meshing point is considered, the calculation accuracy of the meshing point is greatly improved, and the engineering development in the field of face gear meshing performance design and analysis technology is promoted.

[0005] To achieve the above object, the technical solution adopted by the present invention is: a method for calculating the meshing point of a multi-tooth meshing surface gear based on a tooth surface mathematical model, comprising the following steps:

[0006] Step 1: Based on the geometric parameters of the face gear pair, including the number of pinion teeth , Virtual tool teeth number 、Number of teeth on face gear , pressure angle , modulus , axis angle , Inner diameter of face gear , Face gear outer diameter and tooth width , respectively construct the tooth surface mathematical models of the pinion and face gear in the face gear pair, specifically:

[0007] The involute gear is regarded as a virtual tool, the machining process is simulated by the virtual tool, and the tooth surface mathematical model of the face gear is obtained from the tooth surface mathematical model of the virtual tool;

[0008] The specific process of obtaining the mathematical model of the tooth surface of the face gear is: first establish , , and A set of Cartesian coordinate systems, where the coordinate system Origin Located at the center of the virtual tool face circle, coincides with the tool axis, Coincident with the face gear axis; coordinate system Origin yes and The intersection of and Collinear, Parallel to ;

[0009] When the tooth surface mathematical model of the virtual tool generates the tooth surface mathematical model of the face gear, the coordinate system and Fixed, coordinate system and They rotate with the rotation of the virtual tool and the face gear respectively; when the virtual tool rotates around Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle Get; when the gear is around Axis rotation angle , coordinate system By coordinate system Around Axis rotation angle get;

[0010] When generating the mathematical model of the face gear tooth surface, Based on the transformation relationship between the set of Cartesian coordinate systems, the coordinate system To coordinate system The coordinate transformation matrix is ​​defined as:

[0011] ,

[0012] In the formula, is the coordinate origin arrive distance;

[0013] The virtual tool for machining the face gear and the pinion in the face gear pair are both involute tooth surfaces, so:

[0014] Number of teeth on the pinion , modulus and pressure angle , the mathematical model of the pinion tooth surface is constructed by the involute tooth surface equation:

[0015] ,

[0016] ,

[0017] In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; the 1 in the superscript brackets indicates that these vectors are located in the coordinate system , subscript 1 indicates that these parameters belong to the pinion;

[0018] Similarly, the number of virtual tool teeth , modulus and pressure angle , the obtained mathematical model of the tooth surface of the virtual tool is:

[0019] ,

[0020] ,

[0021] In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; Represents these vectors in the coordinate system , subscript It means that these parameters belong to the pinion;

[0022] According to the position vector and normal vector of the tooth surface mathematical model of the virtual tool, the coordinate transformation matrix Obtain the position vector of the mathematical model of the face gear tooth surface for:

[0023] ,

[0024] In the formula is the coordinate transformation matrix mentioned above, Virtual tool winding The rotation angle of the axis;

[0025] At the same time, the position vector The meshing conditions must also be met, then:

[0026] ;

[0027] Similarly, using the coordinate transformation matrix, the normal vector of the mathematical model of the face gear tooth surface is , the normal vector of the virtual tool tooth surface mathematical model Get, that is:

[0028] ,

[0029] In the formula yes The third-order principal form of ;

[0030] Through the geometric parameters of the face gear pair, including the number of pinion teeth , Virtual tool teeth number 、Number of teeth on face gear , pressure angle , modulus , axis angle , Inner diameter of face gear , Face gear outer diameter and tooth width , based on the above process, the mathematical model of the tooth surface of the pinion and the face gear is obtained;

[0031] Step 2: Solve the tooth surface meshing point of tooth pair A:

[0032] Input the mathematical model of the pinion and face gear tooth surface obtained in step 1, and solve the tooth surface meshing point based on the principle of continuous tangential contact; to ensure continuous transmission, the face gear has multiple pairs of teeth meshing at a certain moment, so the following definition is established:

[0033] First, define the tooth pair A as the first to mesh, and the tooth pair B as the second to mesh; before solving the meshing point of the tooth surface, similarly, establish , , and A set of Cartesian coordinates;

[0034] in, and As described in step 1, the coordinate system Origin The center of the circle at the end face of the pinion, Coincident with the axis of the pinion, and parallel;

[0035] Coordinate system As the pinion rotates, it rotates. Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle get;

[0036] Coordinate system where the mathematical model of the pinion tooth surface is located To coordinate system The coordinate transformation matrix The coordinate system of the mathematical model of the gear tooth surface To coordinate system The coordinate transformation matrix , respectively:

[0037] ,

[0038] ,

[0039] In the formula, is the coordinate origin arrive distance;

[0040] Before solving the meshing point, use the coordinate transformation matrix The position vector of the mathematical model of the pinion tooth surface obtained in step 1 is And the law vector Convert to coordinate system , using the coordinate transformation matrix The position vector of the mathematical model of the face gear tooth surface obtained in step 1 is And the law vector Convert to coordinate system ,get,

[0041] ,

[0042] ,

[0043] ,

[0044] ,

[0045] In the formula, and They are and The third-order principal form of ;

[0046] Continuous tangent contact indicates that In this case, the position vectors of the mathematical models of the face gear and pinion tooth surfaces at the meshing point are the same, and the normal vectors are collinear, so there is the following vector equation:

[0047] ,

[0048] ,

[0049] Because the normal vector and The modulus is 1. There are 5 independent scalar equations in the above vector equation, plus the meshing condition , construct a system of 6 equations with 7 unknowns , , , , , and , by inputting The tooth surface meshing point of tooth pair A can be solved by the above vector equations, and different The meshing points corresponding to the different angles of rotation of the pinion gear;

[0050] Step 3: Calculate the angle that the pinion needs to rotate more when tooth pair B enters meshing;

[0051] Due to the influence of manufacturing and installation errors, when tooth pair A enters into meshing, tooth pair B still has tooth surface errors. ; After tooth pair A undergoes elastic deformation and offsets the tooth surface error of tooth pair B, tooth pair B will enter into meshing; Assume that when tooth pair A just enters into meshing, the position vector of the pinion at the meshing point of tooth pair A is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair A and pinion respectively;

[0052] As the pinion rotates around the shaft Rotation, tooth pair A undergoes elastic deformation, and the pinion rotates around the shaft Rotation , the pinion position vector becomes In order for tooth pair B to mesh, tooth pair A needs to produce elastic deformation at the meshing point and tooth surface error of tooth pair B Same, then there is,

[0053] ,

[0054] In addition, from differential geometry we know and Satisfies the following relationship:

[0055] ,

[0056] In the formula, is the coordinate transformation matrix,

[0057] ,

[0058] Combining the above three equations, we can find the angle that the pinion needs to rotate more when the tooth pair B enters meshing. ;

[0059] Step 4: By Angle , the mathematical model of the pinion tooth surface in tooth pair B will be placed in the actual position in the next step to solve the meshing point on tooth pair B taking into account the tooth surface error,

[0060] When tooth pair A just enters meshing, the position vector of the mathematical model of the face gear tooth surface in tooth pair B is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent the tooth pair B and the face gear respectively. The position vector of the mathematical model of the pinion tooth surface is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair B and pinion respectively, and tooth pair A undergoes elastic deformation After that, the pinion rotates around the shaft Turned the corner At this time, in tooth pair B, the position vector of the mathematical model of the pinion tooth surface is given by becomes , then:

[0061] ,

[0062] The pinion gear rotates around the shaft Turned the corner Finally, the mathematical model of the tooth surface of the pinion and face gear in tooth pair B is and , discretize the tooth surfaces of the pinion and face gear along the tooth width and tooth height directions, and mark the tooth width direction of the pinion as , the tooth height direction is marked as ; The tooth width direction of the face gear is marked as , the tooth height direction is marked as ;

[0063] Then the distance from any discrete point on the pinion tooth surface to any discrete point on the face gear tooth surface on tooth pair B is recorded as:

[0064] ,

[0065] Traverse the entire tooth surface, find all distances, and find the point with the smallest distance between the discrete points on the pinion tooth surface and the discrete points on the face gear tooth surface, recorded as:

[0066] ,

[0067] In the above formula is the set of all discrete point distances;

[0068] The pinion angle When , the tooth surface meshing point of tooth pair B is The corresponding discrete points; through the above method, at the pinion angle When the meshing points on tooth pair A and tooth pair B are obtained at the same time.

[0069] Furthermore, after the tooth surface meshing points of the tooth pair A and the tooth pair B are connected, the transmission error and tooth surface contact mark at different meshing points can be obtained, thereby realizing a high-performance load-bearing tooth surface design for the opposing gears.

[0070] The beneficial effects of the present invention are as follows: the present invention provides a method for calculating the meshing point of a multi-tooth meshing face gear based on a tooth surface mathematical model, which takes into account the influence of multi-tooth meshing and tooth surface errors, and can quickly and accurately calculate the tooth surface contact point. At the beginning of gear design, the tooth surface of the face gear can be designed according to different working conditions. The method in this paper not only improves the design efficiency, but also greatly reduces the design cost, and at the same time provides technical support for the design of high-performance transmission system gears. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 is a flow chart of the present invention;

[0072] Figure 2 It is the face gear tooth surface generation coordinate system of the present invention;

[0073] Figure 3 Schematic diagram of the tooth surface structure of the pinion gear of the present invention;

[0074] Figure 4 It is the tooth surface meshing point calculation coordinate system of the present invention;

[0075] Figure 5 is the tooth surface meshing point of the present invention;

[0076] Figure 6 It is a comparison diagram of the tooth surface meshing point errors of the present invention.

[0077] In the figure, 1, virtual tool; 2, face gear; 3, pinion; 4, involute tooth profile; 5, end face; 6, tooth surface meshing point; 7, face gear tooth surface. DETAILED DESCRIPTION

[0078] The principles and features of the present invention are described below in conjunction with the accompanying drawings. The examples given are only used to explain the present invention and are not used to limit the scope of the present invention.

[0079] In order to achieve the above object, the present invention provides the following specific implementation methods: Figure 1 The above is a flowchart of an example implementation. A method for calculating the meshing point of a multi-tooth meshing face gear based on a tooth surface mathematical model comprises the following steps:

[0080] Table 1 Geometric parameters of face gear and pinion

[0081]

[0082] Step 1: Based on the geometric parameters of the face gear and pinion in Table 1, the number of teeth of the pinion 3 、Number of teeth of virtual tool 1 、Number of teeth of face gear 2 , pressure angle , modulus , axis angle , Inner diameter of face gear , Face gear outer diameter , Tooth width , construct the tooth surface equations of the pinion 3 and the face gear 2 in the face gear pair, and obtain the mathematical model of the tooth surface of the face gear pair;

[0083] like Figure 2 As shown in the figure, in order to obtain the mathematical model of the tooth surface of the face gear pair, we first establish , , and A set of Cartesian coordinate systems, where the coordinate system Origin Located at the center of the virtual tool face circle, coincides with the tool axis, Coincident with the face gear axis; coordinate system Origin yes and The intersection of and Collinear, Parallel to ;

[0084] When the tooth surface mathematical model of the virtual tool generates the tooth surface mathematical model of the face gear, the coordinate system and Fixed, coordinate system and They rotate with the rotation of the virtual tool and the face gear respectively; when the virtual tool rotates around Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle Get; when the gear is around Axis rotation angle , coordinate system By coordinate system Around Axis rotation angle get;

[0085] When generating the mathematical model of the face gear tooth surface, Based on the transformation relationship between the set of Cartesian coordinate systems, the coordinate system To coordinate system The coordinate transformation matrix is ​​defined as:

[0086] ,

[0087] In the formula, is the coordinate origin arrive distance;

[0088] The virtual tool for machining the face gear and the pinion in the face gear pair are both involute tooth surfaces, so:

[0089] Number of teeth on the pinion , modulus and pressure angle , the mathematical model of the pinion tooth surface is constructed by the involute tooth surface equation:

[0090] ,

[0091] ,

[0092] In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; the 1 in the superscript brackets indicates that these vectors are located in the coordinate system , subscript 1 means that these parameters belong to the pinion;

[0093] Similarly, the number of virtual tool teeth , modulus and pressure angle , the obtained mathematical model of the tooth surface of the virtual tool is:

[0094] ,

[0095] ,

[0096] In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; Represents these vectors in the coordinate system , subscript It means that these parameters belong to the pinion;

[0097] According to the position vector and normal vector of the tooth surface mathematical model of the virtual tool, the coordinate transformation matrix Obtain the position vector of the mathematical model of the face gear tooth surface for:

[0098] ,

[0099] In the formula is the coordinate transformation matrix mentioned above, Virtual tool winding The rotation angle of the axis;

[0100] At the same time, the position vector The meshing conditions must also be met, then:

[0101] ;

[0102] Similarly, using the coordinate transformation matrix, the normal vector of the mathematical model of the face gear tooth surface is , the normal vector of the virtual tool tooth surface mathematical model Get, that is:

[0103] ,

[0104] In the formula yes The third-order principal form of ;

[0105] Through the geometric parameters of the face gear pair, including the number of pinion teeth , Virtual tool teeth number 、Number of teeth on face gear , pressure angle , modulus , axis angle , Inner diameter of face gear , Face gear outer diameter and tooth width , based on the above process, the mathematical model of the tooth surface of the pinion and the face gear is obtained;

[0106] Step 2: Calculate the tooth surface meshing point of tooth pair A;

[0107] Input the mathematical model of the tooth surface of the pinion and face gear obtained in step 1. Based on the principle of continuous tangential contact, the tooth surface meshing point can be solved. To ensure continuous transmission, the face gear has multiple pairs of teeth meshing at a certain moment. Define the first tooth pair to mesh as tooth pair A, and the subsequent tooth pair to mesh as tooth pair B.

[0108] like Figure 4 As shown, before solving the tooth surface meshing point, similarly, establish , , and A set of Cartesian coordinates;

[0109] in, and As described in step 1, the coordinate system Origin The center of the circle at the end face of the pinion, Coincident with the axis of the pinion, and parallel;

[0110] Coordinate system As the pinion rotates, it rotates. Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle get;

[0111] Coordinate system where the mathematical model of the pinion tooth surface is located To coordinate system The coordinate transformation matrix The coordinate system of the mathematical model of the gear tooth surface To coordinate system The coordinate transformation matrix , respectively:

[0112] ,

[0113] ,

[0114] In the formula, is the coordinate origin arrive distance;

[0115] Before solving the meshing point, use the coordinate transformation matrix The position vector of the mathematical model of the pinion tooth surface obtained in step 1 is And the law vector Convert to coordinate system , using the coordinate transformation matrix The position vector of the mathematical model of the face gear tooth surface obtained in step 1 is And the law vector Convert to coordinate system ,get,

[0116] ,

[0117] ,

[0118] ,

[0119] ,

[0120] In the formula, and They are and The third-order principal form of ;

[0121] Continuous tangent contact indicates that the In this case, the position vectors of the mathematical models of the face gear and pinion tooth surfaces at the meshing point are the same, and the normal vectors are collinear, so there is the following vector equation:

[0122] ,

[0123] ,

[0124] Because the normal vector and The modulus is 1. There are 5 independent scalar equations in the above vector equation, plus the meshing condition , construct a system of 6 equations with 7 unknowns , , , , , and , by inputting The tooth surface meshing point of tooth pair A can be solved by the above vector equations. Figure 5 Shows the different The meshing point corresponding to the value;

[0125] Step 3: Calculate the angle that the pinion needs to rotate more when tooth pair B enters meshing;

[0126] Due to the influence of manufacturing and installation errors, when tooth pair A enters into meshing, tooth pair B still has tooth surface errors. ; After tooth pair A undergoes elastic deformation and offsets the tooth surface error of tooth pair B, tooth pair B will enter into meshing; Assume that when tooth pair A just enters into meshing, the position vector of the pinion at the meshing point of tooth pair A is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair A and pinion respectively;

[0127] As the pinion rotates around the shaft Rotation, tooth pair A undergoes elastic deformation, and the pinion rotates around the shaft Rotation , the pinion position vector becomes In order for tooth pair B to mesh, tooth pair A needs to produce elastic deformation at the meshing point and tooth surface error of tooth pair B Same, then there is,

[0128] ,

[0129] In addition, from differential geometry we know and Satisfies the following relationship:

[0130] ,

[0131] In the formula, is the coordinate transformation matrix,

[0132] ,

[0133] Combining the above three equations, we can find the angle that the pinion needs to rotate more when the tooth pair B enters meshing. ; In this embodiment, the angle that the pinion needs to rotate more is 0.0674 degrees.

[0134] Step 4: By Angle , the mathematical model of the pinion tooth surface in tooth pair B will be placed in the actual position in the next step to solve the meshing point on tooth pair B considering the tooth surface error;

[0135] When tooth pair A just enters meshing, the position vector of the mathematical model of the face gear tooth surface in tooth pair B is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent the tooth pair B and the face gear respectively. The position vector of the mathematical model of the pinion tooth surface is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair B and pinion respectively, and tooth pair A undergoes elastic deformation After that, the pinion rotates around the shaft Turned the corner At this time, in tooth pair B, the position vector of the mathematical model of the pinion tooth surface is given by becomes , then:

[0136] ,

[0137] The pinion rotates around the shaft Turned the corner Finally, the mathematical model of the tooth surface of the pinion and face gear in tooth pair B is and , discretize the tooth surfaces of the pinion and face gear along the tooth width and tooth height directions, and mark the tooth width direction of the pinion as , the tooth height direction is marked as ; The tooth width direction of the face gear is marked as , the tooth height direction is marked as ;

[0138] Then the distance from any discrete point on the pinion tooth surface to any discrete point on the face gear tooth surface on tooth pair B is recorded as:

[0139] ,

[0140] Traverse the entire tooth surface, find all distances, and find the point with the smallest distance between the discrete points on the pinion tooth surface and the discrete points on the face gear tooth surface, recorded as:

[0141] ,

[0142] In the above formula is the set of all discrete point distances;

[0143] The pinion angle When , the tooth surface meshing point of tooth pair B is The corresponding discrete points; through the above method, at the pinion angle When the meshing points on tooth pair A and tooth pair B are obtained at the same time.

[0144] Based on the data in Table 1, it is assumed that the meshing point position vector calculated by the finite element method is , the meshing point position vectors calculated by the proposed method and the traditional method are , , then the meshing point error calculated by this method is defined as,

[0145] ,

[0146] Similarly, the meshing point error calculated by the traditional method can be defined. Figure 6 As shown in the figure, compared with the traditional method, the meshing point error calculated by the proposed method is smaller, and its maximum value is no more than 3%. Therefore, the meshing point calculated by the proposed method is more accurate.

[0147] From the above analysis, it can be seen that when multiple teeth are meshing, due to the influence of tooth surface error, the actual position of the tooth surface meshing point of tooth pair B deviates from the theoretical position. The meshing point calculation method of multi-tooth meshing gears based on the tooth surface mathematical model proposed in this paper can accurately calculate the tooth surface meshing point of tooth pair B by considering the influence of error.

[0148] After the tooth surface meshing points of the tooth pair A and the tooth pair B obtained by the present invention are connected, the transmission error and the tooth surface contact mark at different meshing points can be obtained, thereby realizing the high-performance load-bearing tooth surface design of the opposing gears.

[0149] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for calculating meshing points of multi-tooth meshing surface gears based on a tooth surface mathematical model, characterized in that: The following steps are included: Step 1: Based on the geometric parameters of the face gear pair, including the number of pinion teeth , Virtual tool teeth number 、Number of teeth on face gear , pressure angle , modulus , Axis Angle , Inner diameter of face gear , Face gear outer diameter and tooth width , respectively construct the tooth surface mathematical models of the pinion and face gear in the face gear pair, specifically: The involute gear is regarded as a virtual tool, the machining process is simulated by the virtual tool, and the tooth surface mathematical model of the face gear is obtained from the tooth surface mathematical model of the virtual tool; The specific process of obtaining the mathematical model of the tooth surface of the face gear is: first establish , , and A set of Cartesian coordinate systems, where the coordinate system Origin Located at the center of the virtual tool face circle, coincides with the tool axis, Coincident with the face gear axis; coordinate system Origin yes and The intersection of and Collinear, Parallel to ; When the tooth surface mathematical model of the virtual tool generates the tooth surface mathematical model of the face gear, the coordinate system and Fixed, coordinate system and They rotate with the rotation of the virtual tool and the face gear respectively; when the virtual tool rotates around Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle Get; when the gear is around Axis rotation angle , coordinate system By coordinate system Around Axis rotation angle get; When generating the mathematical model of the face gear tooth surface, Based on the transformation relationship between the set of Cartesian coordinate systems, the coordinate system To coordinate system The coordinate transformation matrix is ​​defined as: ， In the formula, is the coordinate origin arrive distance; The virtual tool for machining the face gear and the pinion in the face gear pair are both involute tooth surfaces, so: Number of teeth on the pinion , modulus and pressure angle , the mathematical model of the pinion tooth surface is constructed by the involute tooth surface equation: ， ， In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; the 1 in the superscript brackets indicates that these vectors are located in the coordinate system , subscript 1 indicates that these parameters belong to the pinion; Similarly, the number of virtual tool teeth , modulus and pressure angle , the obtained mathematical model of the tooth surface of the virtual tool is: ， ， In the formula, is the position vector of the tooth surface mathematical model, is the normal vector of the tooth surface mathematical model, is the tooth profile parameter, is the tooth direction parameter; Represents these vectors in the coordinate system , subscript It means that these parameters belong to the pinion; According to the position vector and normal vector of the tooth surface mathematical model of the virtual tool, the coordinate transformation matrix Obtain the position vector of the mathematical model of the face gear tooth surface for: ， In the formula is the coordinate transformation matrix mentioned above, Virtual tool winding The rotation angle of the axis; At the same time, the position vector The meshing conditions must also be met, then: ； Similarly, using the coordinate transformation matrix, the normal vector of the mathematical model of the face gear tooth surface is , the normal vector of the virtual tool tooth surface mathematical model Get, that is: ， In the formula yes The third-order principal form of ; Through the geometric parameters of the face gear pair, including the number of pinion teeth , Virtual tool teeth number 、Number of teeth on face gear , pressure angle , modulus , axis angle , Inner diameter of face gear , Face gear outer diameter and tooth width , based on the above process, the mathematical model of the tooth surface of the pinion and the face gear is obtained; Step 2: Solve the tooth surface meshing point of tooth pair A: Input the mathematical model of the pinion and face gear tooth surface obtained in step 1, and solve the tooth surface meshing point based on the principle of continuous tangential contact; to ensure continuous transmission, the face gear has multiple pairs of teeth meshing at a certain moment, so the following definition is established: First, define the tooth pair A as the first to mesh, and the tooth pair B as the second to mesh; before solving the meshing point of the tooth surface, similarly, establish , , and A set of Cartesian coordinates; in, and As described in step 1, the coordinate system Origin The center of the circle at the end face of the pinion, Coincident with the axis of the pinion, and parallel; Coordinate system As the pinion rotates, it rotates. Axis rotation angle , coordinate system Also by the coordinate system Around Axis rotation angle get; Coordinate system where the mathematical model of the pinion tooth surface is located To coordinate system The coordinate transformation matrix The coordinate system of the mathematical model of the gear tooth surface To coordinate system The coordinate transformation matrix , respectively: ， ， In the formula, is the coordinate origin arrive distance; Before solving the meshing point, use the coordinate transformation matrix The position vector of the mathematical model of the pinion tooth surface obtained in step 1 is And the law vector Convert to coordinate system , using the coordinate transformation matrix The position vector of the mathematical model of the face gear tooth surface obtained in step 1 And the law vector Convert to coordinate system ,get, ， ， ， ， In the formula, and They are and The third-order principal form of ; Continuous tangent contact indicates that In this case, the position vectors of the mathematical models of the face gear and pinion tooth surfaces at the meshing point are the same, and the normal vectors are collinear, so there is the following vector equation: ， ， Because the normal vector and The modulus is 1. There are 5 independent scalar equations in the above vector equation, plus the meshing condition , construct a system of 6 equations with 7 unknowns , , , , , and , by inputting The tooth surface meshing point of tooth pair A can be solved by the above vector equations, and different The meshing points corresponding to the different angles of rotation of the pinion gear; Step 3: Calculate the angle that the pinion needs to rotate more when tooth pair B enters meshing; Due to the influence of manufacturing and installation errors, when tooth pair A enters into meshing, tooth pair B still has tooth surface errors. ; After tooth pair A undergoes elastic deformation and offsets the tooth surface error of tooth pair B, tooth pair B will enter into meshing; Assume that when tooth pair A just enters into meshing, the position vector of the pinion at the meshing point of tooth pair A is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair A and pinion respectively; As the pinion rotates around the shaft Rotation, tooth pair A undergoes elastic deformation, and the pinion rotates around the axis Rotation , the pinion position vector becomes In order for tooth pair B to mesh, tooth pair A needs to produce elastic deformation at the meshing point and tooth surface error of tooth pair B Same, then there is, ， In addition, from differential geometry we know and Satisfies the following relationship: ， In the formula, is the coordinate transformation matrix, ， Combining the above three equations, we can find the angle that the pinion needs to rotate more when the tooth pair B enters meshing. ; Step 4: By Angle , the mathematical model of the pinion tooth surface in tooth pair B will be placed in the actual position in the next step to solve the meshing point on tooth pair B taking into account the tooth surface error, When tooth pair A just enters meshing, the position vector of the mathematical model of the face gear tooth surface in tooth pair B is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent the tooth pair B and the face gear respectively. The position vector of the mathematical model of the pinion tooth surface is , the superscript 1 in the brackets represents the position vector in the coordinate system , subscript and 1 represent tooth pair B and pinion respectively, and tooth pair A undergoes elastic deformation After that, the pinion rotates around the shaft Turned the corner At this time, in tooth pair B, the position vector of the mathematical model of the pinion tooth surface is given by becomes , then: ， The pinion gear rotates around the shaft Turned the corner Finally, the mathematical model of the tooth surface of the pinion and face gear in tooth pair B is and , discretize the tooth surfaces of the pinion and face gear along the tooth width and tooth height directions, and mark the tooth width direction of the pinion as , the tooth height direction is marked as ; The tooth width direction of the face gear is marked as , the tooth height direction is marked as ; Then the distance from any discrete point on the pinion tooth surface to any discrete point on the face gear tooth surface on tooth pair B is recorded as: ， Traverse the entire tooth surface, find all distances, and find the point with the smallest distance between the discrete points on the pinion tooth surface and the discrete points on the face gear tooth surface, recorded as: ， In the above formula is the set of distances of all discrete points; The pinion angle When , the tooth surface meshing point of tooth pair B is The corresponding discrete points; through the above method, at the pinion angle When the meshing points on tooth pair A and tooth pair B are obtained at the same time.

2. A method for calculating meshing points of multi-tooth meshing surface gears based on a tooth surface mathematical model as claimed in claim 1, characterized in that: After the tooth surface meshing points of the tooth pair A and the tooth pair B are connected, the transmission error and the tooth surface contact mark at different meshing points can be obtained, thereby realizing the high-performance load-bearing tooth surface design of the opposite gears.

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