A method for forming a ball gear by milling

Abstract

The present application belongs to the technical field of gear machining, and discloses a forming milling machining method of spherical gear. The method uses a disc-shaped milling cutter to machine the spherical gear, and the steps are as follows: firstly, the parameters of the spherical gear are given, the tooth surface equation of the spherical gear is derived, and the tooth surface normal vector is calculated; secondly, the forming milling machining coordinate system of the spherical gear is established, and the homogeneous transformation matrix between the coordinate systems is derived, and then the contact line equation between the spherical gear and the disc-shaped milling cutter is derived; then, the cross-sectional profile of the disc-shaped milling cutter is calculated, and then the rotary surface of the disc-shaped milling cutter is obtained; finally, the spherical gear is formed and milled by using the calculated disc-shaped milling cutter, and after machining an annular gear slot, the disc-shaped milling cutter is rotated by a certain angle to continue machining the next gear slot. The forming milling machining method of the spherical gear proposed by the present application can realize the machining of the spherical gear without relying on special gear machining machine tools, and the disc-shaped milling cutter used has a simple structure and is 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 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, traditional gear processing methods are difficult to directly apply to ball gear processing.

[0003] Invention patent CN 101406974 B discloses a device for machining spherical gears, which can realize the machining 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 achieve the generating grinding of spherical gears.

[0004] A review of existing literature reveals that existing ball gear machining methods fall into two main categories. The first involves forming using finger-shaped tools, but this method suffers from a low material removal rate and results in low machining efficiency. The second involves generating ball gears based on the principle of gear meshing. However, this method places high demands on the machine tool structure and results in a complex tool structure, making tool manufacturing inconvenient. Given the shortcomings of existing ball gear machining technology, it is necessary to propose a ball gear machining method that both ensures machining efficiency and facilitates 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 this 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 headspace coefficient

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

[0010]

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

[0012]

[0013] The tooth profile curve of the spherical gear rotates around its polar axis to obtain the annular tooth surface of the spherical gear, 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 processing 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 aligned with 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 z t x t y t z t z

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

[0022]

[0023] where a is the center distance of the disc cutter installation, and Σ is the angle between the disc cutter axis and the polar axis of the spherical gear (i.e., the angle between z w and z t );

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

[0025]

[0026] Step three, derive the contact line equation between the spherical gear and the disc cutter;

[0027] According to the gear forming method processing principle, in the forming milling process, the spherical gear and the disc cutter need to satisfy the conjugate contact condition shown in the following formula:

[0028] n t ·v 12 = 0

[0029] where n t has been calculated in step two, and v 12 can be expressed as:

[0030]

[0031] After sorting, the conjugate contact condition of the spherical gear forming milling is:

[0032]

[0033] Then the point on the spatial contact line in the workpiece coordinate system S w satisfies the following equation:

[0034]

[0035] Step four, calculate the disc cutter cross-section profile, and then get the disc cutter rotary surface;​

[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 machining one 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: This invention addresses the milling process for ball gears, proposes a milling method for ball gears using a disc milling cutter, and provides specific embodiments. The milling method can be performed on a general-purpose five-axis CNC machine tool, eliminating the need for specialized gear processing machines. Furthermore, the disc milling cutter used is simple in structure and easy to manufacture. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0045] Figure 2 The 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 aligned with 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 axis of the disc milling cutter, 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 milling cutter;

[0048] Figure 4 The three-dimensional model of the disc milling cutter, where (a) is the side view and (b) is the main 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] The following takes the forming milling of concave ball gears as an example to explain a forming milling method of ball gears based on a disc milling cutter. 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 headspace coefficient The given ball gear parameters are shown in Table 1;

[0052] Table 1 Parameters of the 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 follows 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 processing 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] 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 (i.e. z w Axis and z t The installation center distance a=40mm and the included 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 three, deduce the contact line equation between the globoidal gear and the disc cutter;

[0070] According to the principle of gear forming method, the globoidal gear and the disc cutter should satisfy the conjugate contact condition shown in the following equation during the forming milling process:

[0071] n t ·v 12 =0

[0072] Wherein, n t has been calculated in step two, v 12 can be expressed as:

[0073]

[0074] The conjugate contact condition of globoidal gear forming milling can be obtained as follows:

[0075]

[0076] The points on the spatial contact line in the workpiece coordinate system S w satisfy the following equation:

[0077]

[0078] Step four, calculate the disc cutter cross-section profile, and then get the disc cutter rotary surface;

[0079] A series of discrete points on the spatial contact line in the workpiece coordinate system S w have been calculated in step three, represented as {P i =(P xi ,P yi ,P zi ,1)′,i=0,1,2,3…}, according to the homogeneous transformation matrix in step two, the discrete point set {P i} is converted to the cutter coordinate system S t :

[0080]

[0081] In the cutter coordinate system S t , the contact line discrete point set is projected to the disc cutter cross-section (i.e. the x t O t z t plane of the cutter coordinate system S t ), represented as

[0082]

[0083] Wherein,

[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 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 and establish a three-dimensional model of the disc milling cutter, 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 machining one 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 present invention proposes a ball gear forming milling method based on a disc milling cutter, which is not limited to ball gears with an involute tooth profile curve, 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 merely 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 scope of protection 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 , number of teeth , pressure angle , displacement coefficient , tooth addendum coefficient and headspace coefficient ; Step 2: Establish the ball gear forming milling processing coordinate system, including the workpiece coordinate system and tool coordinate system , and derive the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system; Step 3, derive the contact line equation between the ball gear and the disc milling cutter; Step 4: Calculate the cross-sectional profile of the disc milling cutter, and then obtain 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 machining an annular tooth groove, the disc milling cutter moves around the workpiece coordinate system. of Axis rotation angle , continue processing 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 spherical gear parameters, in the spherical gear base circle plane, the involute tooth profile curve of the spherical gear is expressed as: ； in, is the right tooth profile, The left tooth profile line; is the expansion angle of any point on the involute; is the base circle radius of the ball gear, is the half angle of the ball gear tooth groove, and 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: ； in, is the rotation angle of the point on the involute around the polar axis, parameter and The value ranges are: ； According to the tooth surface equation of the spherical gear, 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 The coordinate origin Located at the center of the ball gear, The shaft coincides with the polar axis of the ball gear. The shaft is in the plane of the ball gear base circle and Axis vertical, Axis and axis, The axes form a right-handed Cartesian coordinate system; the tool coordinate system The coordinate origin Located in the center of the disc milling cutter, The axis coincides with the axis of the disc milling cutter. axis, The axis is within the cross section of the disc milling cutter and The axes form a right-handed Cartesian coordinate system; According to the workpiece coordinate system With tool coordinate system The relative position relationship, the homogeneous transformation matrix from the workpiece coordinate system to the tool coordinate system Expressed as: ； in, The center distance for the disc milling cutter installation, is the angle between the disc milling cutter axis and the ball gear polar axis. Axis and The angle between the axes; Then the point and normal vector on the tooth surface of the spherical gear are in the tool coordinate system In Chinese it is represented as: ； 。 4. The ball gear forming milling method according to claim 3, characterized in that: The specific implementation process of step three is: According to the principle of gear forming, during the forming milling process, the ball gear and the disc milling cutter must meet the conjugate contact conditions shown in the following formula: ； in, Already calculated in step 2, Expressed as: ； The conjugate contact conditions for ball gear forming milling can be obtained as follows: ； Then the workpiece coordinate system 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 four is: The workpiece coordinate system has been calculated in step 3 A series of discrete points on the contact line in space are expressed as , according to the homogeneous transformation matrix in step 2, the discrete point set Convert to tool coordinate system : ； In the tool coordinate system In the contact line discrete point set Projected onto the cross section of the disc milling cutter, i.e. the tool coordinate system of Surface, expressed as : ； in, ; Tool coordinate system 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 wound around The axis rotates to obtain the rotating surface of the disc milling cutter.

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