Face gear pair meshing equation initial value searching and solving method

Abstract

The invention discloses a face gear pair meshing equation initial value searching and solving method which is used for calculating an iterative calculation initial value of a face gear pair meshing equation so as to accurately solve a face gear pair contact trace, and through face gear pair tooth surface discrete solving, the process that a face gear pair tooth surface enters meshing is simulated, and a tooth surface discrete point cloud is globally searched and solved. Determining an initial calculation value of the meshing equation set; the method improves the calculation efficiency and the calculation precision, provides theoretical support for the contact characteristic research of the face gear pair, and fills up the international related technical blank.

Description

Technical Field

[0001] The invention relates to the technical field of gear transmission, and in particular to an initial value search and solution method for a face gear pair meshing equation. Background Art

[0002] Face gears have broad application prospects in the field of aerospace transmission due to their unique advantages such as light configuration and low sensitivity to installation. The contact trace is an indispensable calculation parameter for studying the contact characteristics of face gear pairs. However, due to the nonlinear characteristics of the face gear meshing equation, it is computationally difficult to accurately and quickly solve it. For nonlinear equations, calculating the initial value is crucial to solving the equation.

[0003] In order to solve the above problems, the present invention proposes a method for searching and solving the initial values ​​of the meshing equations of face gear pairs. By discretely solving the tooth surfaces of the face gear pairs, the process of the tooth surfaces of the face gear pairs entering meshing is simulated, and a global search and solution is performed on the discrete point cloud of the tooth surfaces, thereby determining the calculation initial values ​​of the meshing equation group, thereby improving the calculation efficiency and reliability, providing theoretical support for the study of the contact characteristics of face gear pairs, filling the gap in related international technologies, and promoting the development of engineering technology. At the same time, it has great social and economic benefits. Summary of the invention

[0004] In order to overcome the deficiencies of the prior art and fill the gaps in the relevant technology, the present invention provides a method for searching for an initial value of a meshing equation of a face gear pair. The technical solution adopted by the present invention to solve the technical problem is as follows:

[0005] A method for searching for an initial value of a meshing equation of a face gear pair, characterized by comprising the following steps:

[0006] Step (1): Set the parameters of the face gear pair, construct the face gear pair tooth surface forming coordinate system, and derive the gear coordinate system S according to the face gear forming mechanism. s To the face gear coordinate system S b The coordinate transformation matrix M bs , the position vector r of the gear is obtained by s With the normal vector n s Tooth surface equation generated by envelope r g With position vector n g :

[0007]

[0008] Where L bs The coordinate transformation matrix M bs The 3×3 matrix obtained by removing the homogeneous coordinates;

[0009] Step (2): According to the relative position relationship of the face gear pair when meshing, construct the face gear contact analysis coordinate system, and derive the coordinate transformation matrix M from the cylindrical gear coordinate system and the face gear coordinate system to the contact coordinate system. cp and M cg :

[0010]

[0011] in, is the angle that the cylindrical gear rotates in the contact coordinate system; is the angle that the face gear rotates in the contact coordinate system;

[0012] The contact coordinate system S is obtained after the face gear coordinate transformation a The tooth surface equation of the face gear in g (a) and the normal vector n g (a) , and the cylindrical gear tooth surface equation r p (a) and the normal vector n p (a) , combined to get the meshing equation of the face gear pair:

[0013]

[0014] in, is the angle that the cylindrical gear rotates in the contact coordinate system; w p is the involute parameter of cylindrical gear; u p Indicates the tooth parameters of cylindrical gears; is the angle that the face gear rotates in the contact coordinate system; is the angle that the generating gear rotates in the generating motion coordinate system; w s is the involute parameter of the gear;

[0015] The meshing equation of the face gear pair is a set of equations containing 6 unknown quantities and 5 independent nonlinear equations. Considered as known quantities, the meshing equation can be transformed into a nonlinear equation system containing 1 known quantity and 5 unknown quantities. To solve;

[0016] Step (3): Calculate the tooth surfaces of the face gear and the cylindrical gear, and calculate the tooth surfaces of the face gear and the cylindrical gear in the tooth width direction and the tooth height direction. a and i b The number is discretized to obtain the discrete point cloud data of the face gear and cylindrical gear tooth surface i a ×i b points;

[0017] Step (4): Set the contact judgment convergence value Δ1 and initialize the cylindrical gear rotation angle Rotation angle with face gear 0, the first cycle calculation starts, set the cylindrical gear rotation angle Incremental step size h, Assign values ​​so that Reset the gear tooth surface positions i and j to 0, start the loop calculation, add 1 to i, and determine whether i satisfies i≤i a +1 condition, if the condition is not met, reset i to 0, if the condition is met, add 1 to j to determine whether j satisfies j≤i b +1 condition. If the condition is not met, reset j to 0, and add 1 to i, and repeat the cycle again. If the condition is met, reset the face gear tooth surface position m and n to 0, add 1 to m, and judge whether m satisfies i≤i a +1 condition, if the condition is not met, reset m to 0, and add 1 to j; if the condition is met, add 1 to n to determine whether n satisfies n≤i b +1 condition. If the condition is not met, reset n to 0 and add 1 to m. If the condition is met, record the discrete tooth surface point p of the cylindrical gear. ij Coordinate value (x p ,y p ,z p ) and the discrete tooth contact points p of the face gear mn Coordinate value (x g ,y g ,z g );

[0018] Step (5): Rotate the cylindrical gear by the calculated angle Rotation angle with face gear Substitute the coordinate transformation matrix M cp and M cg , calculate the discrete tooth surface point p of the cylindrical gear ij Discrete tooth surface contact point p of the face gear mn The coordinate value distance D pg :

[0019]

[0020] Among them, r1(x1,y1,z1) is the coordinate vector of the cylindrical gear in the contact coordinate system after rotation; r2(x2,y2,z2) is the coordinate vector of the face gear in the contact coordinate system;

[0021] Determine the coordinate value distance D pg Does it meet condition D? pg≤Δ1, if the condition is met, the calculation is terminated and the point cloud position coordinates of the cylindrical gear and the face gear at this time are output as the meshing starting point, and the rotation angle value of the cylindrical gear at this time is recorded As the initial value of the meshing equation If the condition is not met, n is assigned a value of 1, and the coordinate value distance D at the new position is calculated again pg Repeat the tooth surface search calculation until the search of all discrete point clouds of the cylindrical gear tooth surface against the discrete point cloud of the opposite gear at the current rotation angle is completed. If all are not satisfied, the cylindrical gear rotation angle is calculated. Re-assign Similarly, the next round of tooth surface search begins until the coordinate value distance D is met. pg <Δ1, the search for the initial value of the meshing equation can be completed;

[0022] Step (6): The initial value of the tooth surface obtained by searching and solving Substitute the meshing equation into the nonlinear equation system The solution calculation is performed in:

[0023]

[0024] Substitution The discrete point p on the cylindrical gear tooth surface corresponding to the time ij Coordinate value (x p ,y p ,z p ) and the discrete tooth surface contact point p of the face gear mn Coordinate value (x g ,y g ,z g ), the nonlinear equations can be eliminated and the initial value of the tooth surface As the angle rotation value of the cylindrical gear pair at the meshing point, calculate the disengagement time of the face gear pair The contact trajectory formed by the rigid contact on the tooth surface during the meshing-in to meshing-out process of the face gear pair can be extracted by using the angular rotation value of the cylindrical gear pair at the meshing-out point. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a flow chart for implementing the initial value search and solution method of the meshing equation of a face gear pair proposed by the present invention;

[0026] Figure 2 It is a schematic diagram of coordinate transformation from the shape gear coordinate system to the face gear coordinate system of the present invention;

[0027] Figure 3 It is a schematic diagram of the discrete point cloud search of the face gear pair of the present invention;

[0028] Figure 4 It is a schematic diagram of the distance of the point cloud coordinate values ​​of the face gear pair of the present invention;

[0029] Figure 5 is a schematic diagram of the contact trace of the cylindrical gear of the present invention;

[0030] Figure 6 It is a schematic diagram of the contact trace of the face gear of the present invention. DETAILED DESCRIPTION

[0031] The embodiments of the present invention are described with reference to the accompanying drawings. Figure 1 — Figure 6 The specific embodiments of the present invention are described in detail.

[0032] Reference Figure 1 , a method for searching for an initial value of a meshing equation of a face gear pair, comprising the following steps:

[0033] Step (1): Set the parameters of the face gear pair and construct Figure 2 The face gear pair tooth surface forming coordinate system shown in the figure, the moving coordinate system S s -X s Y s Z s The coordinate system of the gear is fixedly connected to the gear coordinate system. The moving coordinate system S in the figure is b -X b Y b Z b Fixedly connected to the face gear coordinate system, the auxiliary coordinate system S m -X m Y m Z m and S n -X n Y n Z n It is fixed to the frame. During the shaping motion, the shaping gear rotates around the axis by an angle of w. s , the face gear rotates around the axis by an angle of w b , generating gear position vector r s With the normal vector n s It can be expressed as:

[0034]

[0035] Among them, w 0s is the involute parameter of the gear, u s Represents the gear tooth parameters.

[0036] From the gear coordinate system S s To the face gear coordinate system S b The coordinate transformation matrix M bs It can be expressed as:

[0037]

[0038] in, is the angle that the generating gear rotates in the generating motion coordinate system; is the angle that the face gear rotates in the generating motion coordinate system;

[0039] Face gear position vector r g and the normal vector n g The cylindrical gear position vector r s and the normal vector n s After coordinate transformation, it can be expressed as:

[0040]

[0041] Where L bs The coordinate transformation matrix M bs The 3×3 matrix obtained by removing the homogeneous coordinates;

[0042] Step (2): Figure 2 As shown in the figure, according to the relative position relationship of the face gear pair when meshing, the face gear contact analysis coordinate system is constructed. In order to ensure the point contact of the face gear pair, the cylindrical gear in the face gear pair should have 1-3 fewer teeth than the shape gear. The cylindrical gear position vector r p With the normal vector n p Similar to the shape of the gear. p -X p Y p Z p To contact coordinate system S c -X c Y c Z c The coordinate transformation matrix M cp It can be expressed as:

[0043]

[0044] in, is the angle that the cylindrical gear rotates in the contact coordinate system;

[0045] From the face gear coordinate system S g -X g Y g Z g To contact coordinate system S c -X c Y c Z c The coordinate transformation matrix M cg It can be expressed as:

[0046]

[0047] in, is the angle that the face gear rotates in the contact coordinate system;

[0048] After the cylindrical gear coordinates are transformed, the contact coordinate system S a The tooth surface equation in p (a) and the normal vector n p (a) It can be expressed as;

[0049]

[0050] Where L cp The coordinate transformation matrix M cp The 3×3 matrix obtained by removing the homogeneous coordinates;

[0051] After the face gear coordinates are transformed, they are in the contact coordinate system S a The tooth surface equation in g (a) and the normal vector n g (a) It can be expressed as:

[0052]

[0053] Where L cg The coordinate transformation matrix M cg The 3×3 matrix obtained by removing the homogeneous coordinates;

[0054] Combining the above formulas, the cylindrical gear and the face gear can be placed in the contact coordinate system S a The meshing equation in can be expressed as:

[0055]

[0056] Among them, w p is the involute parameter of cylindrical gear; u p Indicates the tooth parameters of cylindrical gears;

[0057] Step (3): If Figure 3 As shown in the figure, the calculated tooth surfaces of the face gear and cylindrical gear are solved respectively, and the tooth width direction and tooth height direction are calculated by i a and i b The number is discretized to obtain the discrete point cloud data of the tooth surface of the face gear pair i a ×i b points;

[0058] Step (4): Set the contact judgment convergence value Δ1 and initialize the cylindrical gear rotation angle Rotation angle with face gear 0, the first cycle calculation starts, set the cylindrical gear rotation angle Incremental step size h, Assign values ​​so that Reset the gear tooth surface positions i and j to 0, start the loop calculation, add 1 to i, and determine whether i satisfies i≤i a +1 condition, if the condition is not met, reset i to 0, if the condition is met, add 1 to j to determine whether j satisfies j≤i b +1 condition. If the condition is not met, reset j to 0, and add 1 to i, and repeat the cycle again. If the condition is met, reset the face gear tooth surface position m and n to 0, add 1 to m, and judge whether m satisfies i≤i a +1 condition, if the condition is not met, reset m to 0, and add 1 to j; if the condition is met, add 1 to n to determine whether n satisfies n≤i b +1 condition. If the condition is not met, reset n to 0 and add 1 to m. If the condition is met, record the discrete tooth surface point p of the cylindrical gear. ij Coordinate value (x p ,y p ,z p ) and the discrete tooth contact points p of the face gear mn Coordinate value (x g ,y g ,z g );

[0059] Step (5): Figure 4 As shown, the cylindrical gear rotation angle calculated by the cycle is Rotation angle with face gear Substitute the coordinate transformation matrix M cp and M cg , calculate the discrete tooth surface point p of the cylindrical gear ij Discrete tooth surface contact point p of the face gear mn The coordinate value distance D pg :

[0060]

[0061] Among them, r1(x1,y1,z1) is the coordinate vector of the cylindrical gear in the contact coordinate system after rotation; r2(x2,y2,z2) is the coordinate vector of the face gear in the contact coordinate system;

[0062] Determine the coordinate value distance D pg Does it meet condition D? pg ≤Δ1, if the condition is met, the calculation is terminated and the point cloud position coordinates of the cylindrical gear and the face gear at this time are output as the meshing starting point, and the rotation angle value of the cylindrical gear at this time is recorded As the initial value of the meshing equation If the condition is not met, n is assigned a value of 1, and the coordinate value distance D at the new position is calculated again pg Repeat the tooth surface search calculation until the search of all discrete point clouds of the cylindrical gear tooth surface against the discrete point cloud of the opposite gear at the current rotation angle is completed. If all are not satisfied, the cylindrical gear rotation angle is calculated. Re-assign Similarly, the next round of tooth surface search begins until the coordinate value distance D is met. pg <Δ1, the search for the initial value of the meshing equation can be completed;

[0063] Step (6): The initial value of the tooth surface obtained by searching and solving Substitute the meshing equation into the nonlinear equation system The solution calculation is performed in:

[0064]

[0065] Substitution The discrete point p on the cylindrical gear tooth surface corresponding to the time ij Coordinate value (x p ,y p ,z p ) and the discrete tooth surface contact point p of the face gear mn Coordinate value (x g ,y g ,z g ), the nonlinear equations can be eliminated and the initial value of the tooth surface As the angle rotation value of the cylindrical gear pair at the meshing point, calculate the disengagement time of the face gear pair As the angle rotation value of the cylindrical gear pair at the meshing point, the contact trajectory formed by the rigid contact on the tooth surface of the face gear pair from meshing in to meshing out can be extracted. The schematic diagram of the contact trajectory of the cylindrical gear and the face gear tooth surface calculated is shown in Figure 5 and Figure 6 shown. Figure 5 The horizontal coordinate is the axial coordinate value of the cylindrical gear, the vertical coordinate is the radial coordinate value of the cylindrical gear, and the track line on the tooth surface is the track line of the cylindrical gear from meshing in to meshing out during the meshing process; Figure 6 The horizontal coordinate is the radial coordinate value of the face gear, and the vertical coordinate is the axial coordinate value of the face gear. Similarly, the trajectory line on the tooth surface is the trajectory of the face gear from meshing in to meshing out during the meshing process.

[0066] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modification, change and equivalent changes made to the above embodiment according to the essence of the invention shall still fall within the protection scope of the technology of the invention.

Claims

1. A method for searching for initial values ​​of meshing equations of face gear pairs, characterized by: Step (1): Set the face gear pair parameters, construct the face gear pair tooth surface forming coordinate system, and obtain the gear position vector r s With the normal vector n s Tooth surface equation generated by envelope r g With position vector n g : Step (2): Based on the relative position relationship of the face gear pair when meshing, derive the coordinate transformation matrix M from the cylindrical gear coordinate system and the face gear coordinate system to the contact coordinate system. cp and M cg ; Get the contact coordinate system S after the face gear coordinate transformation a The tooth surface equation of the face gear in g (a) and the normal vector n g (a) , and the cylindrical gear tooth surface equation r p (a) and the normal vector n p (a) , combined to get the meshing equation of the face gear pair; Step (3): Calculate the tooth surfaces of the face gear and the cylindrical gear, and calculate the tooth surfaces of the face gear and the cylindrical gear in the tooth width direction and the tooth height direction. a and i b The number is discretized to obtain the discrete point cloud data of the face gear and cylindrical gear tooth surface i a ×i b points; Step (4): Set the contact judgment convergence value Δ1 and initialize the cylindrical gear rotation angle Rotation angle with face gear 0, the first cycle calculation starts, set the cylindrical gear rotation angle Incremental step size h, for Assign values ​​so that Reset the gear tooth surface positions i and j to 0, start the loop calculation, add 1 to i, and determine whether i satisfies i≤i a +1 condition, if the condition is not met, reset i to 0, if the condition is met, add 1 to j to determine whether j satisfies j≤i b +1 condition. If the condition is not met, reset j to 0, and add 1 to i, and repeat the cycle again. If the condition is met, reset the face gear tooth surface position m and n to 0, add 1 to m, and judge whether m satisfies i≤i a +1 condition, if the condition is not met, reset m to 0, and add 1 to j; if the condition is met, add 1 to n to determine whether n satisfies n≤i b +1 condition. If the condition is not met, reset n to 0 and add 1 to m. If the condition is met, record the discrete tooth surface point p of the cylindrical gear. ij Coordinate value (x p ,y p ,z p ) and the discrete tooth contact points p of the face gear mn Coordinate value (x g ,y g ,z g ); Step (5): Rotate the cylindrical gear by the calculated angle Rotation angle with face gear Substitute the coordinate transformation matrix M cp and M cg , calculate the discrete tooth surface point p of the cylindrical gear ij Discrete tooth surface contact point p of the face gear mn The coordinate value distance D pg : Among them, r1(x1,y1,z1) is the coordinate vector of the cylindrical gear in the contact coordinate system after rotation; r2(x2,y2,z2) is the coordinate vector of the face gear in the contact coordinate system; Determine the coordinate value distance D pg Does it meet condition D? pg ≤Δ1, if the condition is met, the calculation is terminated and the point cloud position coordinates of the cylindrical gear and the face gear at this time are output as the meshing starting point, and the rotation angle value of the cylindrical gear at this time is recorded As the initial value of the meshing equation If the condition is not met, n is assigned a value of 1, and the coordinate value distance D at the new position is calculated again pg Repeat the tooth surface search calculation until the search of all discrete point clouds of the cylindrical gear tooth surface against the discrete point cloud of the opposite gear at the current rotation angle is completed. If all are not satisfied, the cylindrical gear rotation angle is calculated. Re-assign Similarly, the next round of tooth surface search begins until the coordinate value distance D is met. pg <Δ1, the search for the initial value of the meshing equation can be completed; Step (6): The initial value of the tooth surface obtained by searching and solving By substituting it into the meshing equation and solving it, the contact trajectory line formed on the tooth surface during the meshing to meshing process of the face gear pair can be extracted.

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