Spherical gear forming and milling method

Abstract

The invention belongs to the technical field of gear machining, and discloses a ball gear forming and milling method. The method uses a disc-shaped milling cutter to process the ball gear, and comprises the following steps of: firstly, giving parameters of the ball gear, deducing a tooth surface equation of the ball gear and calculating a tooth surface normal vector; secondly, a spherical gear forming and milling coordinate system is established, a homogeneous transformation matrix between the coordinate systems is derived, and then a contact line equation between the spherical gear and the disc-shaped milling cutter is derived; then, calculating the cross section profile of the disc-shaped milling cutter, and further obtaining a rotary surface of the disc-shaped milling cutter; and finally, the calculated disc-shaped milling cutter is used for conducting forming milling machining on the ball gear, and after one annular tooth groove is machined, the disc-shaped milling cutter rotates by a certain angle to continue to machine the next tooth groove. According to the spherical gear forming and milling machining method, machining of the spherical gear can be achieved without depending on a special machining tool for the gear, and the used disc-shaped milling cutter is simple in structure and easy to manufacture.

Description

Technical Field

[0001] The invention belongs to the technical field of gear processing and relates to a ball gear forming milling processing method. Background Art

[0002] The ball gear pair is a new type of dual-degree-of-freedom gear transmission mechanism. Compared with traditional gear mechanisms such as cylindrical gears and bevel gears, there are significant differences in the tooth surface shape. Therefore, the traditional gear processing method is difficult to directly apply to ball gear processing.

[0003] Invention patent CN 101406974 B discloses a device for processing spherical gears, which can realize the processing of spherical gears using a finger-shaped milling cutter on a universal lifting table milling machine; document "006-2343 (2008) 04-073-03" proposes a design scheme for a spherical gear forming grinding machine tool, in which a finger-shaped grinding wheel is used as a grinding tool; document "10.1016 / j.mechmachtheory.2009.03.005" introduces a spherical gear grinding method, which uses a toothed disc-shaped grinding wheel to realize the generating grinding of spherical gears.

[0004] By consulting the existing literature, it can be seen that the existing ball gear processing methods are mainly divided into two categories. The first category is to use finger-shaped tools for forming processing, but this method has a low material removal rate, resulting in low processing efficiency; the second category is based on the principle of gear meshing, and the ball gear is processed by the expansion method, but this method has high requirements on the machine tool structure, and the tool structure is complex, which is not convenient for tool manufacturing. Based on the shortcomings of the existing ball gear processing technology, it is necessary to propose a ball gear processing method that can both ensure processing efficiency and facilitate tool manufacturing. Summary of the invention

[0005] The present invention proposes a ball gear forming milling processing method based on a disc milling cutter. The core of the method is to achieve efficient and high-precision forming milling processing of the ball gear by designing a disc milling cutter that accurately matches the tooth profile of the ball gear to be processed.

[0006] The technical solution of the present invention:

[0007] A ball gear forming milling method based on a disc milling cutter, the steps are as follows:

[0008] Step 1: Given the spherical gear parameters, derive the spherical gear tooth surface equation and calculate the tooth surface normal vector; the spherical gear parameters include the module m n , number of teeth z, pressure angle α n , displacement coefficient x, tooth top height coefficient and head clearance coefficient

[0009] According to the given ball gear parameters, in the ball gear base circle plane, the involute tooth profile curve of the ball gear can be expressed as:

[0010]

[0011] Among them, i=1 is the right tooth profile line, and i=2 is the left tooth profile line; is the development angle of any point on the involute; r b is the base circle radius of the ball gear, σ0 is the half angle of the tooth groove of the ball gear, r b And σ0 can be calculated according to the following formula:

[0012]

[0013] The spherical gear tooth profile curve rotates around its polar axis to obtain the spherical gear annular tooth surface, which can be expressed as:

[0014]

[0015] Among them, θ is the rotation angle of the point on the involute around the polar axis, and the parameter The value ranges of and θ are:

[0016]

[0017] According to the spherical gear tooth surface equation, the tooth surface normal vector can be expressed as:

[0018]

[0019] Step 2: Establish the ball gear forming milling machining coordinate system, including the workpiece coordinate system S w (O w -x w y w z w ) and tool coordinate system S t (Q t -x t y t z t ), and derive the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system;

[0020] Among them, the workpiece coordinate system S w The coordinate origin O w Located at the center of the ball gear, z w The axis coincides with the polar axis of the ball gear, x w The axis is in the plane of the ball gear base circle and is parallel to the z w Axis vertical, y w Axis and x w Axis, z w The axes form a right-handed Cartesian coordinate system; the tool coordinate system S t The coordinate origin Ot Located in the center of the disc milling cutter, z t The axis coincides with the disc milling cutter axis, x t Axis, y t The axis is within the cross section of the disc milling cutter and is aligned with the z t The axes form a right-handed Cartesian coordinate system;

[0021] According to the workpiece coordinate system S w With tool coordinate system S t The relative position relationship, the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system It can be expressed as:

[0022]

[0023] Among them, a is the installation center distance of the disc milling cutter, Σ is the angle between the axis of the disc milling cutter and the polar axis of the ball gear (i.e. z w Axis and z t The angle between the axes);

[0024] Then the point and normal vector on the tooth surface of the spherical gear are in the tool coordinate system S t It can be expressed as:

[0025]

[0026] Step 3, deriving the contact line equation between the ball gear and the disc milling cutter;

[0027] According to the principle of gear forming method, during the forming milling process, the ball gear and the disc milling cutter must meet the conjugate contact conditions shown in the following formula:

[0028] n t ·v 12 =0

[0029] Among them, n t It has been calculated in step 2, v 12 It can be expressed as:

[0030]

[0031] After sorting, the conjugate contact conditions for ball gear forming milling are:

[0032]

[0033] Then the workpiece coordinate system S w The points on the mid-space contact line satisfy the following equation:

[0034]

[0035] Step 4, calculating the cross-sectional profile of the disc milling cutter, and then obtaining the rotary surface of the disc milling cutter;

[0036] The workpiece coordinate system S has been calculated in step 3 w A series of discrete points on the contact line in the space are represented by {P i =(P xi ,P yi ,P zi ,1)′,i=0,1,2,3…}, according to the homogeneous transformation matrix in step 2, the discrete point set {P i}Transform to tool coordinate system S t :

[0037]

[0038] In the tool coordinate system S t In the contact line discrete point set Projected onto the cross section of the disc milling cutter (i.e. tool coordinate system S t x t O t z t ), expressed as

[0039]

[0040] in,

[0041] Tool coordinate system x t O t z t A set of discrete points within a surface Fitting is performed to obtain the cross-sectional profile of the disc milling cutter; finally, the cross-sectional profile of the disc milling cutter is revolved around z t The axis rotates to obtain the rotating surface of the disc milling cutter;

[0042] Step 5: Use the disc milling cutter calculated in step 4 to perform forming milling on the ball gear. After processing an annular tooth groove, the disc milling cutter moves around the workpiece coordinate system S w of y w The axis rotates by an angle of 2π / z and continues to process the next annular tooth groove.

[0043] Beneficial effects of the invention: The invention is oriented to the ball gear forming milling process, proposes a ball gear forming milling method based on a disc milling cutter, and gives a specific embodiment. The ball gear forming milling method proposed by the invention can be implemented on a general five-axis CNC machine tool without relying on a gear-specific processing machine tool, and the disc milling cutter used has a simple structure and is easy to manufacture. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a three-dimensional model of ball gear;

[0045] Figure 2 Coordinate system for ball gear forming milling;

[0046] Among them, S w (O w -x w y w z w ) is the workpiece coordinate system, the coordinate origin O w Located at the center of the ball gear, z w The axis coincides with the polar axis of the ball gear, x w The axis is in the plane of the ball gear base circle and is parallel to the z w Axis vertical, y w Axis and x w Axis, z w The axes form a right-handed Cartesian coordinate system; S t (O t -x t y t z t ) is the tool coordinate system, the coordinate origin O t Located in the center of the disc milling cutter, z t The axis coincides with the disc milling cutter axis, x t Axis, y t The axis is within the cross section of the disc milling cutter and is aligned with the z t The axes form a right-handed Cartesian coordinate system;

[0047] Figure 3 The cross-sectional profile of the disc-shaped milling cutter;

[0048] Figure 4 It is a three-dimensional model of a disc milling cutter, where (a) is a side view and (b) is a front view. DETAILED DESCRIPTION

[0049] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0050] Taking the forming milling of concave ball gears as an example, a ball gear forming milling method based on a disc milling cutter is described below. The specific steps are as follows:

[0051] Step 1: Given the spherical gear parameters, derive the spherical gear tooth surface equation and calculate the tooth surface normal vector; the spherical gear parameters include the module m n , number of teeth z, pressure angle α n , displacement coefficient x, tooth top height coefficient and head clearance coefficient The given ball gear parameters are shown in Table 1;

[0052] Table 1 Parameters of ball gear to be processed

[0053]

[0054] According to the given ball gear parameters, the annular tooth surface of the ball gear can be expressed as:

[0055]

[0056] Among them, according to the technical solution, r b and σ0 are:

[0057] r b =30.07mm

[0058] σ0=1.96°

[0059] The value ranges of and θ are:

[0060]

[0061] By solving the spherical gear tooth surface equation and performing 3D modeling in CAM software, the spherical gear 3D model is obtained as shown in the figure. Figure 1 As shown;

[0062] According to the spherical gear tooth surface equation, the tooth surface normal vector can be expressed as:

[0063]

[0064] Step 2: Establish the ball gear forming milling machining coordinate system, such as Figure 2 As shown, including the workpiece coordinate system S w (O w -x w y w z w ) and tool coordinate system S t (O t -x t y t z t ), and derive the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system:

[0065]

[0066] Where a is the installation center distance of the disc milling cutter, ∑ is the angle between the axis of the disc milling cutter and the polar axis of the ball gear (i.e. z w Axis and z t The installation center distance a=40mm and the angle ∑=22.5° are given;

[0067] Then the point and normal vector on the tooth surface of the spherical gear are in the tool coordinate system S t It can be expressed as:

[0068]

[0069] Step 3, deriving the contact line equation between the ball gear and the disc milling cutter;

[0070] According to the principle of gear forming method, during the forming milling process, the ball gear and the disc milling cutter must meet the conjugate contact conditions shown in the following formula:

[0071] n t ·v 12 =0

[0072] Among them, n t Already calculated in step 2, v 12 It can be expressed as:

[0073]

[0074] After sorting, the conjugate contact conditions for ball gear forming milling are:

[0075]

[0076] Then the workpiece coordinate system S w The points on the mid-space contact line satisfy the following equation:

[0077]

[0078] Step 4, calculating the cross-sectional profile of the disc milling cutter, and then obtaining the rotary surface of the disc milling cutter;

[0079] The workpiece coordinate system S has been calculated in step 3 w A series of discrete points on the contact line in the space are represented by {P i =(P xi ,P yi ,P zi ,1)′,i=0,1,2,3…}, according to the homogeneous transformation matrix in step 2, the discrete point set {P i}Transform to tool coordinate system S t :

[0080]

[0081] In the tool coordinate system S t In the contact line discrete point set Projected onto the cross section of the disc milling cutter (i.e. tool coordinate system S t x t O t z t ), expressed as

[0082]

[0083] in,

[0084] Discrete point set The coordinates of some points in are shown in Table 2:

[0085] Table 2 Tool coordinate system x t O t z t Coordinates of some discrete points in the surface

[0086]

[0087]

[0088] Tool coordinate system x t O t z t A set of discrete points within a surface Fitting is performed to obtain the cross-sectional profile of the disc milling cutter, such as Figure 3 Finally, the disc milling cutter cross-section profile is revolved around z t The axis rotates to obtain the rotating surface of the disc milling cutter, and a three-dimensional model of the disc milling cutter is established, such as Figure 4 As shown;

[0089] Step 5: Use the disc milling cutter calculated in step 4 to perform forming milling on the ball gear. After processing an annular tooth groove, the disc milling cutter moves around the workpiece coordinate system S w of y w The axis rotates 11.25° and continues to process the next annular tooth groove.

[0090] The ball gear forming milling processing method based on a disc milling cutter proposed in the present invention is not limited to ball gears whose tooth profile curve is an involute, but is also applicable to ball gears with arc and other curved tooth profiles.

[0091] It will be easily understood by those skilled in the art that 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 and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A ball gear forming milling method, characterized in that: Here are the steps: Step 1: Given the spherical gear parameters, derive the spherical gear tooth surface equation and calculate the tooth surface normal vector; the spherical gear parameters include the module m n , number of teeth z, pressure angle α n , displacement coefficient x, tooth top height coefficient and head clearance coefficient Step 2: Establish the ball gear forming milling machining coordinate system, including the workpiece coordinate system S w (O w -x w y w z w ) and tool coordinate system S t (O t -x t y t z t ), and derive the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system; Step 3, deriving the contact line equation between the ball gear and the disc milling cutter; Step 4, calculating the cross-sectional profile of the disc milling cutter, and then obtaining the rotary surface of the disc milling cutter; Step 5: Use the disc milling cutter calculated in step 4 to perform forming milling on the ball gear. After processing an annular tooth groove, the disc milling cutter moves around the workpiece coordinate system S w of y w The axis rotates by an angle of 2π / z and continues to process the next annular tooth groove.

2. The ball gear forming milling method according to claim 1, characterized in that: The specific implementation process of step one is: According to the given ball gear parameters, in the ball gear base circle plane, the involute tooth profile curve of the ball gear is expressed as: Among them, i=1 is the right tooth profile line, and i=2 is the left tooth profile line; is the development angle of any point on the involute; r b is the base circle radius of the ball gear, σ0 is the half angle of the tooth groove of the ball gear, r b And σ0 is calculated according to the following formula: The tooth profile curve of the spherical gear rotates around its polar axis to obtain the annular tooth surface of the spherical gear, which is expressed as: Among them, θ is the rotation angle of the point on the involute around the polar axis, and the parameter The value ranges of and θ are: According to the spherical gear tooth surface equation, the tooth surface normal vector is expressed as:

3. The ball gear forming milling method according to claim 1, characterized in that: The specific implementation process of step 2 is: Workpiece coordinate system S w The coordinate origin O w Located at the center of the ball gear, z w The axis coincides with the polar axis of the ball gear, x w The axis is in the plane of the ball gear base circle and is parallel to the z w Axis vertical, y w Axis and x w Axis, z w The axes form a right-handed Cartesian coordinate system; the tool coordinate system S t The coordinate origin O t Located in the center of the disc milling cutter, z t The axis coincides with the disc milling cutter axis, x t Axis, y t The axis is within the cross section of the disc milling cutter and is aligned with the z t The axes form a right-handed Cartesian coordinate system; According to the workpiece coordinate system S w With tool coordinate system S t The relative position relationship, the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system It is expressed as: Among them, a is the installation center distance of the disc milling cutter, ∑ is the angle between the axis of the disc milling cutter and the polar axis of the ball gear, that is, z w Axis and z t The angle of the axes; Then the point and normal vector on the tooth surface of the spherical gear are in the tool coordinate system S t In the expression:

4. The ball gear forming milling method according to claim 1, characterized in that: The specific implementation process of step three is: According to the principle of gear forming method, during the forming milling process, the ball gear and the disc milling cutter must meet the conjugate contact conditions shown in the following formula: n t ·v 12 =0 Among them, n t It has been calculated in step 2, v 12 It is expressed as: After sorting, the conjugate contact conditions for ball gear forming milling are: Then the workpiece coordinate system S w The points on the mid-space contact line satisfy the following equation:

5. The ball gear forming milling method according to claim 4, characterized in that: The specific implementation process of step 4 is as follows: The workpiece coordinate system S has been calculated in step 3 w A series of discrete points on the contact line in the space are represented by {P i =(P xi ,P yi ,P zi ,1)′,i=0,1,2,3…}, according to the homogeneous transformation matrix in step 2, the discrete point set {P i }Transform to tool coordinate system S t : In the tool coordinate system S t In the contact line discrete point set Projected onto the cross section of the disc milling cutter, i.e. the tool coordinate system S t x t O t z t Surface, expressed as in, Tool coordinate system x t O t z t A set of discrete points within a surface Fitting is performed to obtain the cross-sectional profile of the disc milling cutter; finally, the cross-sectional profile of the disc milling cutter is revolved around z t The axis rotates to obtain the rotating surface of the disc milling cutter.

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