Method for calculating cross-ball distance of asymmetric inner and outer inverted conical teeth

Abstract

The invention discloses a method for calculating the cross-ball distance of asymmetric inner and outer back taper teeth. The method comprises the following steps of: 1, solving basic parameters of a workpiece, wherein the basic parameters comprise a base circle diameter, an original circle division end surface pressure angle and actual circle division helical angles on the left and right sides of an inclined back taper; 2, deducing the cross-ball distance at the position L of any cross section of the tooth pitch large end of the outer inclined back taper: solving the tooth thickness of the original large end face of the outer inclined back taper, the arc tooth thickness of the end face of any T cross section, the equivalent original normal arc tooth thickness of the T cross section, the equivalent original circle dividing normal displacement coefficient, the involute angle of the equivalent ball measuring center circle pressure angle of the T cross section and the ball measuring center circle pressure angle; and finally, calculating the cross-ball distance of the cross section of the odd and even teeth of the outer inclined inverted conical teeth at any distance L from the large end. According to the method, a solution is given, a calculation formula of the solution is deduced and the technical problem is solved by aiming at the difficulty of the cross-ball distance of the inner and outer inclined inverted conical teeth of the large gear ring, a theoretical calculation method of the cross-ball distance of the inverted conical teeth is perfected through three-dimensional modeling verification that the error is extremely small, and the method has guiding significance on field processing and detection of the inclined inverted conical teeth.

Description

Technical Field

[0001] This invention belongs to the field of gear design technology, specifically relating to a method for calculating the ball spacing of asymmetric internal and external inverted bevel teeth. Background Technology

[0002] Compared to a straight inverted cone, an oblique inverted cone is an asymmetrical inverted cone. Unlike helical teeth, the left and right helical surfaces of an oblique inverted cone are asymmetrical, meaning the actual helical surfaces on the left and right sides of the same cylindrical surface are different, while the end face tooth profile parameters and base circle parameters are the same. The left and right helical surfaces of a helical tooth are completely identical, and its ball contact point lies on two parallel helical lines of the same contact circle. The ball-span distance of a helical tooth can be calculated using formulas (38~43) in Section 6.3.2 of the national standard GB / Z 18620.2-2008.

[0003] Since the left and right helical surfaces of the oblique conical tooth are different, the left and right contact points are on two non-parallel helical lines of the same distribution circle. Although the national standard clearly defines the calculation method for the span of the ball of the oblique tooth, it does not consider the oblique conical tooth.

[0004] Currently, most inverted bevel teeth on the market are straight inverted bevel teeth. Internally inclined inverted bevel teeth are mainly achieved through scraping and rotary milling; externally inclined inverted bevel teeth are mainly achieved through scraping and hobbing. Furthermore, there is no relevant literature that clearly derives the calculation of the ball-span distance of inclined inverted bevel teeth. Summary of the Invention

[0005] In view of the above-mentioned shortcomings and defects of the prior art, the purpose of this invention is to provide a method for calculating the span of an asymmetric inner and outer inverted conical tooth, thereby solving the problem that there is no method for calculating the span of an oblique inverted conical tooth in the prior art.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: a method for calculating the span of an asymmetric inner and outer conical tooth: The distance between the ball and the outer bevel tooth is calculated using the following formula:

[0007]

[0008] The ball spacing of the inverted bevel gear is calculated using the following formula:

[0009]

[0010] in: When the number of teeth is even, the external oblique bevel tooth spans the ball distance. When the number of teeth is odd, the external oblique bevel tooth spans the ball distance. When the number of teeth is even, the distance between the inward bevel teeth and the ball is... When the number of teeth is odd, the span of the internal oblique bevel tooth across the ball is... : Measuring the diameter of a sphere Original end face pressure angle. : Modulus of the normal direction of the circle, Pi (π) The original large end's normal tooth thickness at the biting circle. The original large end of the internal tooth has a normal arc groove. Original circle-split helix angle, : Actual helix angle on the left side of the dividing circle The actual helix angle on the right side of the dividing circle. Base circle diameter; Number of teeth.

[0011] Compared with the prior art, the beneficial technical effects of this invention are: (I) This invention provides a solution to the difficulty of the ball spacing of the inner and outer oblique conical teeth of the large gear ring, derives its calculation formula, solves this technical problem, and verifies that the error is minimal after three-dimensional modeling. It improves the theoretical calculation method of the ball spacing of the conical teeth and has guiding significance for the on-site processing and inspection of oblique conical teeth. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of an inwardly oblique conical tooth.

[0013] Figure 2 This is a schematic diagram of the outward-curved conical tooth segmentation.

[0014] Figure 3 This is a schematic diagram of the unfolded shape of a straight inverted conical sphere in contact with a cylinder.

[0015] Figure 4 This is a schematic diagram of the helical gear contacting the measuring sphere in the measuring plane as part of the cylinder.

[0016] Figure 5 A schematic diagram showing the unfolding of a cylindrical contact ball within the same measuring plane of an inclined cone.

[0017] Figure 6 Calculation of tooth thickness at any T-section end face of the external left oblique conical tooth.

[0018] Figure 7 Calculation of tooth thickness at any T-section end face of the inner left oblique conical tooth (circle expansion).

[0019] Figure 8 Example verification of UG simulation Figure I .

[0020] Figure 9 Example verification of UG simulation Figure II .

[0021] The specific content of the present invention will be further explained in detail below with reference to the embodiments. Detailed Implementation

[0022] It should be noted that, unless otherwise specified, all raw materials used in this invention are those known in the prior art.

[0023] The following are specific embodiments of the present invention. It should be noted that the present invention is not limited to the following specific embodiments. All equivalent modifications made based on the technical solutions of this application fall within the protection scope of the present invention.

[0024] This invention analyzes the left and right helical surfaces and the contact characteristics of the measuring ball of straight-reverse cone, helical tooth, and oblique-reverse cone teeth as follows: Within the same measuring plane, the contact points of the measuring sphere on the left and right helical surfaces of the straight and inverted cones lie on symmetrical, opposite, equivalent helical lines along the same contact circle, such as... Figure 3 It belongs to the normal tangential contact, with no axial misalignment between the left and right contact points, and different contact circles on different measuring surfaces, with gradual changes in tooth thickness, and different center circles and span distances of the measuring sphere. Within the same measuring plane, the contact points of the measuring sphere on the left and right helical surfaces of the helical teeth lie on parallel helical lines of the same contact circle, such as... Figure 4 It belongs to the normal tangent point contact, the left and right contact points are axially misaligned, and the contact circles of different measuring surfaces are the same, the tooth thickness is the same, and the center circle of the measuring ball and the span of the ball are the same. Within the same measuring plane, the contact points of the measuring sphere on the left and right helical surfaces of the inclined cone are located on non-parallel helical lines of the same contact circle, such as... Figure 5 It also belongs to the normal tangent point contact, with the left and right contact points axially misaligned. However, the tooth thickness of different measuring surfaces gradually changes, resulting in different contact circles, and the center circle of the measuring ball and the distance between the balls are also different.

[0025] This invention proposes a method for calculating the distance between the oblique inverted cone and the sphere: First, the accurate calculation formula for the end face tooth thickness of the oblique inverted cone at any cross section is derived; Secondly, determining the span distance necessitates calculating the pressure angle of the measuring sphere's center circle. The pressure angle of the measuring sphere's center circle at the normal tangent contact point must be converted to end-face parameters for calculation. While the left and right normal tooth profile parameters of the oblique cone differ, the left and right end-face parameters are identical and equal to the end-face parameters of its original helical tooth. Therefore, calculating the pressure angle of the measuring sphere's center circle on the end face of the oblique cone is equivalent to calculating the pressure angle of the measuring sphere's center circle under the end-face parameters of its original helical tooth; simply substitute the precise tooth thickness of the measurement section. Finally, the precise end face tooth thickness of the oblique cone at the measurement surface needs to be calculated and converted into the equivalent normal tooth thickness of the original oblique tooth, and then its span-ball distance can be solved.

[0026] Most inverted bevel gears on the market are straight inverted bevel gears, such as... Figure 1 Internally tapered teeth are mainly achieved through scraping and rotary milling; externally tapered teeth, such as... Figure 2 This is mainly achieved through tooth scraping and hobbing, and there is no clear derivation in the literature for calculating the ball spacing of the oblique bevel teeth.

[0027] Example 1: A method for calculating the span of an asymmetric inner and outer conical tooth sphere includes the following steps: Step 1: Determine the basic parameters of the workpiece: Base circle diameter:

[0028] Original end face pressure angle:

[0029] The actual pitch circle helix angles on both sides of the oblique inverted cone

[0030]

[0031] Where: z: number of teeth on the workpiece. Normal pressure angle of the bisection circle, : Pressure angle of the end face of the circle Original circle-split helix angle, : Left and right circular spiral increment, : Actual helix angle on the left side of the dividing circle : Actual helix angle on the right side of the dividing circle; Step 2: Derive the span-sphere distance at any cross section L of the large end of the external oblique conical tooth pitch. The original large end face tooth thickness of the externally oblique inverted cone, such as Figure 6 :

[0032] : Original large end bifurcation normal tooth thickness Arbitrary T-section end face arc tooth thickness:

[0033] L : Axial distance from the large end to the T-face Since L=BM, then That is:

[0034]

[0035] have:

[0036]

[0037] Equivalent original normal arc tooth thickness of T-section:

[0038]

[0039] Equivalent original bicircle normal displacement coefficient:

[0040]

[0041]

[0042] Substituting the equivalent original normal parameters of section T into formulas (38~43) in section 6.3.2 of GB / Z 18620.2-2008, the involute angle of the pressure angle of the center circle of the equivalent mass sphere at section T is calculated. :

[0043]

[0044] in ; Further use the general inverse involute function Find the pressure angle at the center of the unique sphere. :

[0045]

[0046] Finally, the span-sphere distance of the cross section at any distance L from the large end of the odd and even numbered teeth of the external oblique conical tooth is calculated as follows:

[0047]

[0048] have:

[0049]

[0050] Step 3: Solve for the spherical distance at any cross section L of the inner oblique inverted cone at the large end. Similarly, the end face arc groove of the thicker end of the internal tooth, i.e., the thinner end of the tooth groove, is as follows: Figure 7 :

[0051] Arbitrary T-section end face arc tooth groove: , Right now:

[0052]

[0053] The width of the arc groove on the end face of the internal oblique inverted conical tooth can be derived from any cross-section. : have:

[0054]

[0055] Furthermore, the equivalent original normal arc groove at any cross section L :

[0056] Equivalent original bicircle normal displacement coefficient:

[0057]

[0058]

[0059] Similarly, the distance between the inverted cone and the ball of the internal tooth is:

[0060]

[0061] in ; Further use the inverse involute function Find the pressure angle at the center circle of the sphere. :

[0062]

[0063] Finally, the span-sphere distance at any distance L from the tooth thickness end of the odd and even numbered teeth of the internal oblique conical tooth is calculated as follows:

[0064]

[0065] have:

[0066]

[0067] in: When the number of teeth is even, the external oblique bevel tooth spans the ball distance. When the number of teeth is odd, the external oblique bevel tooth spans the ball distance. When the number of teeth is even, the distance between the inward bevel teeth and the ball is... When the number of teeth is odd, the span of the internal oblique bevel tooth across the ball is... : Measuring the diameter of a sphere Original end face pressure angle. : Modulus of the normal direction of the circle, Pi (π) The original large end's normal tooth thickness at the biting circle. The original large end of the internal tooth has a normal arc groove. Original circle-split helix angle, : Actual helix angle on the left side of the dividing circle The actual helix angle on the right side of the dividing circle. Base circle diameter; Number of teeth.

[0068] Example verification: Internal oblique bevel tooth large end normal parameters 77, 2.964 22° 11° 4.7338, left and right side chamfer angles ±3.25°, calculate the ball span distance at L=4mm from the large end, and measure the ball. 4mm. The verification process for this patent is as follows: Based on the above, first obtain the sub-circle on the left side of the groove. The right side of the groove is divided into circles. ; Pressure angle at the end face of the circle ; base circle ; The end face tooth groove width of the oblique bevel tooth at L=4mm is calculated as follows:

[0069] Next, the equivalent normal tooth groove width of the bevel tooth at L=4mm is calculated. as follows:

[0070]

[0071] Substituting further, we obtain the distance M between the inner oblique inverted cone and the sphere at the L=4mm section:

[0072] like Figure 8As shown, the span distance obtained by precise 3D modeling of the aforementioned inner oblique conical teeth using UG is 230.5611 mm, with an error of 0.02 mm compared to the formula calculation. This error is fully in line with engineering applications, as shown in the figure below: fully in line with engineering applications.

[0073] The normal parameter z of the large end of the externally inclined conical tooth is 72. 3.5, 20° 13°L, 7.0202, left and right side chamfer angles ±3.25°, calculate the ball span distance at L=4mm from the large end, and measure the ball. 5mm. The verification process for this patent is as follows: Based on the above, the basic parameters of the outward-sloping inverted cone are first obtained: Helix angle on the left side of the tooth thickness pitch circle , Helix angle on the right side of the tooth thickness pitch circle ; Large end split end face pressure angle ; base circle ; The end face arc thickness of the external oblique conical tooth at L=4mm is calculated as follows: ; Next, the equivalent normal arc thickness of the beveled tooth at L=4mm is calculated. as follows: ; Substituting further, we obtain the span distance M of the outer oblique inverted cone at the L=4mm section:

[0074]

[0075] like Figure 9 As shown, the span distance M=265.99 obtained by precise 3D modeling of the above-mentioned external oblique conical teeth using UG is 0.02mm, which is completely consistent with the engineering application.

[0076] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions conceived by those skilled in the art within the scope of the technology disclosed in the present invention without creative effort are covered within the scope of protection of the present invention.

Claims

1. A method for calculating the cross-pitch of asymmetric inner and outer involute conical gears, characterized in that, the cross-pitch of outer inclined involute conical gears is calculated according to the following formula: the cross-pitch of inner inclined involute conical gears is calculated according to the following formula: wherein: : Transverse ball pitch of external oblique involute gear with even number of teeth : Transverse ball pitch of outer oblique involute gear with odd number of teeth : inner oblique inverted cone gear with even number of teeth, span of balls : inner oblique inverted cone gear with odd number of teeth, span of ball : measure ball diameter; : original involute end face pressure angle; : cyclotomic method to modulus; : pi; : original big end involute tooth thickness; : inner tooth original large end circular method normal arc tooth slot; : original cyclotorsional angle; : actual helix angle left side of circle : actual helix angle right side of circle : base circle diameter; : number of teeth.

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